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On the admissibility of scale and quantile estimators Brewster, John Frederick 1972

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ON THE ADMISSIBILITY OF SCALE AND QUAJSITILE ESTIMATORS  BY c  JOHN FREDERICK BREWSTER B.Sc, University of B r i t i s h Columbia, 1966 M.Sc., University of Toronto, 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS  We accept this thesis as conforming to the required  standard  The University of B r i t i s h Columbia March 1972  In  presenting  an  advanced  the I  Library  further  for  degree shall  agree  scholarly  by  his  of  this  written  this  thesis  in  at  University  the  make  that  it  purposes  for  may  t  e  It  financial  of  for  is  A p r i l 28. 1972  of  of  Columbia,  British  by  the  understood  gain  Columbia  for  extensive  granted  Mathematics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  a  be  fulfilment  available  permission.  Department  D  freely  permission  representatives. thesis  partial  shall  Head o f  be  requirements  reference copying  that  not  the  and  of my  I agree  this  or  allowed  without  that  study. thesis  Department  copying  for  or  publication my  ii  Supervisor:  James V. Zidek  ABSTRACT  The i n a d m i s s i b i l i t y of the best a f f i n e - i n v a r i a n t estimators for the variance and noncentral quantiles of the normal law, when loss i s squared error, has already been established.  However, the proposed (minimax)  alternatives to the usual (minimax, but inadmissible) estimators are themselves inadmissible. In our search for admissible a l t e r n a t i v e s i n these problems, we f i r s t consider estimators which are formal Bayes within the class of scale-invariant procedures.  For such estimators, we present  explicit  conditions for a d m i s s i b i l i t y within the class cf s c a l e - i n v a r i a n t procedures . In the second chapter of the thesis, we consider the estimation of an a r b i t r a r y power of the scale parameter of a normal population.  Under  the assumption that the loss function s a t i s f i e s c e r t a i n reasonable cond i t i o n s , an estimator i s constructed which i s (i) minimax, and  ( i i ) for-  mal Bayes within the class of s c a l e - i n v a r i a n t procedures.  estimator  The  obtained i s a l i m i t of a sequence of minimax, preliminary test estimators. Moreover, under squared  error loss, and using the r e s u l t s of Chapter  this estimator i s shown to be scale-admissible.  One,  More generally, r e s u l t s  are obtained for the estimation of powers of the scale parameter i n the canonical form of the general l i n e a r model, and for the estimation of powers of the scale parameter of an exponential d i s t r i b u t i o n with unknown  iii  location. In Chapter Three, conditions are given f o r the minimaxity variant procedures  i n general location-scale problems.  of best i n -  F i n a l l y , by com-  bining these r e s u l t s with those of the preceding chapter, the usual i n t e r v a l estimators f o r the variance of a normal population are shown to be minimax, but inadmissible.  Superior, minimax procedures  are suggested.  iv  TABLE OF CONTENTS  page INTRODUCTION  1  CHAPTER ONE.: Scale-admissible, Invariant Estimators For Quantiles And Variance Of The Normal Law Under Quadratic Loss  4  1.1  Introduction And Summary  4  1.2  D e f i n i t i o n s And Preliminary Results  4  1.3  The Reduced Problem  9  1.4  Quantile Estimation  14  1.5  Scale Estimation  39  CHAPTER TWO: Minimax, Scale-admissible Estimators Of Scale Parameters  42  2.1  Introduction  42  2.2  Inadmissibility Of The Best  2.3  The Construction Of A Minimax, Formal Bayes Scale-Invariant Estimator  2.4  Extension Of The Previous Results To Loss Functions  2.5  The Exponential D i s t r i b u t i o n With Unknown Location And Scale  56  CHAPTER THREE: Minimax, Inadmissible Interval Estimators Of Scale Parameters  60  G-invariant Estimator  G-invariant  43 48' 53  3.1  Minimax Estimators In Location-Scale Problems  60  3.2  I n t e r v a l Estimation Of Scale Parameters  70  BIBLIOGRAPHY  "  78  V  ACKNOWLEDGEMENT  I am deeply indebted to Professor James Zidek for h i s suggestion of the problems treated i n this thesis and for his guidance during i t s preparation.  I am p a r t i c u l a r l y thankful to him for the knowledge and insight  he has imparted  to me during the course of my graduate program.  I would also l i k e to extend my appreciation to Professors Lawrence Clevenson, Ned G l i c k , Stanley Nash, and C a r l Sarndal for their c a r e f u l reading of this d i s s e r t a t i o n and for their h e l p f u l comments concerning it.  In p a r t i c u l a r , I would l i k e to thank Professor Richard Shorrock for  suggestions given during the preparation of the second chapter. F i n a l l y , I would l i k e to thank Eleanor Lannon for her care and patience i n typing the  manuscript.  The f i n a n c i a l support of the U n i v e r s i t y of B r i t i s h Columbia and the National Research Council of Canada i s g r a t e f u l l y acknowledged.  Chapter 0:  Introduction  From the point of view of decision theory, the s t a t i s t i c i a n ' s job consists of the s e l e c t i o n of a decision r u l e from the large class of rules a v a i l a b l e .  To make his task easier, the s t a t i s t i c i a n often imposes  r e s t r i c t i o n s which reduce, i n s i z e , the class of a v a i l a b l e r u l e s .  As  many s t a t i s t i c a l problems possess c e r t a i n natural symmetries, the " p r i n c i p l e of invariance"  i s often employed i n this manner.  Although there  i s r a r e l y a "best" decision rule i n a p a r t i c u l a r problem, there may  be  a best r u l e among the invariant ones.  im-  posing the p r i n c i p l e of invariance rules to  In such cases, the e f f e c t of  i s to reduce the class of a l l possible  one.  I t may  have seemed, at one  time, that the best invariant r u l e would,  except i n unusual circumstances, have most of the properties desirable i n decision theory.  considered  Such a b e l i e f would have been supported by  the work of K i e f e r [22] and Kudo [23], who  showed that the best invariant  r u l e i s minimax i n a wide class of problems.  Also, i t i s w e l l known that  the best invariant r u l e i s admissible i f the group acting on the problem i s compact.  In 1956,  however, Stein [36] showed that the usual  (best  a f f i n e - invariant) estimator for the multivariate normal mean i s inadmissible under squared error loss i f the dimension i s greater than or equal to 3.  Brown [4] extended Stein's r e s u l t to a very general location  parameter problem i n which he found that the dichotomy between 3 or more dimensions and less than 3 dimensions p e r s i s t s . Stein [37] also showed that the usual estimator (again, best a f f i n e invariant) f o r the variance of a normal population i s inadmissible  under  2  squared error loss i f the mean i s unknown.  Brown [5]  also extended this  r e s u l t to a wider class of d i s t r i b u t i o n s and loss functions.  In addition  to the above r e s u l t s , Zidek [46] has shown that the usual estimator for any non-central quantile of a normal d i s t r i b u t i o n i s inadmissible. A common feature of the aforementioned  papers i s that the proposed  (minimax) a l t e r n a t i v e s to the usual (minimax, but inadmissible) estimators are themselves inadmissible.  Strawderman [38] and Brown [6] have recently  presented admissible, minimax estimators for the multivariate normal mean. As the usual estimators for the variance and non-central quantiles of the normal law are inadmissible, i t i s natural to search for admissible a l t e r n a t i v e s i n these problems, too.  From many points of view i t i s  reasonable to r e s t r i c t this search to the class of formal Bayes estimators.  For, l i k e proper Bayes estimators, formal Bayes estimators are  comparatively  easy to obtain i n an e x p l i c i t form for many commonly used  loss functions.  Furthermore, i t i s w e l l known that Bayes estimators  and  their l i m i t s , i n an appropriate topology, constitute a complete class. And i n many i n t e r e s t i n g s t a t i s t i c a l problems (see Sacks [32], F a r r e l l [11], and Brown [6]) these l i m i t s are formal Bayes estimators.  In the  f i r s t chapter of the thesis, e x p l i c i t conditions are given for the s c a l e a d m i s s i b i l i t y of formal Bayes, s c a l e - i n v a r i a n t estimators for the variance and quantiles of the normal law, under squared error l o s s .  These conditions  are obtained by a p p l i c a t i o n of an extension of the results of Zidek [45]. The second chapter i s concerned primarily with the estimation of an a r b i t r a r y power of the scale parameter of a normal population.  Under the  assumption that the loss function s a t i s f i e s certain reasonable conditions, an estimator i s constructed which i s ( i ) minimax, and  ( i i ) formal Bayes  3  within the class of s c a l e - i n v a r i a n t procedures.  The estimator obtained  i a a l i m i t of a sequence of minimax, preliminary test estimators, each constructed i n the manner of Brown.  Moreover, under squared  using the r e s u l t s of Chapter One,  this estimator i s shown to be s c a l e -  admissible.  error l o s s , and  More generally, results are obtained for the estimation of  powers of the scale parameter i n the canonical form of-the general l i n e a r model, and f o r the estimation of powers of the scale parameter of an exponential d i s t r i b u t i o n with unknown l o c a t i o n . The remainder of the thesis i s devoted p r i m a r i l y to i n t e r v a l estimation problems.  F i r s t , conditions are given for the minimaxity  best i n v a r i a n t procedures  i n general location-scale problems.  of  As applied  to confidence i n t e r v a l s , the main theorem may be viewed as an extension of Valand  [40].  F i n a l l y , the usual i n t e r v a l estimators f o r the variance  of a normal population are shown to be minimax, but inadmissible. minimax procedures  are  suggested.  Superior,  4  Chapter 1:  Scale-admissible, Invariant Estimators For Quantiles  Variance Of The Normal Law Under Quadratic  1.1  And  Loss  Introduction And Summary The usual estimators f o r the variance and noncentral quantiles of  the normal d i s t r i b u t i o n are known to be inadmissible, under squared error loss ([37], [46]). suggested.  However, no admissible a l t e r n a t i v e s have been  This chapter gives e x p l i c i t conditions f o r the  scale-admis-  s i b i l i t y , under squared error l o s s , of formal Bayes, s c a l e - i n v a r i a n t estimators  <$>^  and  ty^  f°  r  W + no  0  and  M  , respectively.  If  0*  rep-  resents the orbit space created by the action of the scale group on the parameter space, and i f  ty  on  IT ( i = l , 2) , then, under s u i t a b l e r e g u l a r i t y con-  9*  with density  ditions,  ty.  i s formal Bayes with respect-to a p r i o r measure  i s scale-admissible i f  1  =  00  .  " 2 -1 f(t Tr.(t)) dt 1 1  _ 1  =  f(jt  2  Tr..(t))  -1  dt  _oo  These conditions are obtained by a p p l i c a t i o n of an extension of  the r e s u l t s of Zidek [45]. 1.2  D e f i n i t i o n s And Preliminary Results Let  (X, 8) .  X  be a random v a r i a b l e taking i t s values i n a measurable space  Assume that the d i s t r i b u t i o n of  X  i s a unique but unknown mem-  ber of a family of p r o b a b i l i t y d i s t r i b u t i o n s indexed by a set  0 , a  (possibly unbounded) subinterval of the r e a l l i n e , with upper and lower endpoints  and  A f t e r observing  8^ , r e s p e c t i v e l y . X , a real-valued (measurable) function  i s to be estimated, with a loss function, L: form  w:  0 x X -*• (R  A x 0 x X -> [0, °°) , of the.  5  L(a, 6, x) = c(9, x) || a - w(6, x ) | | ,  (1.2.1)  2  where the action space, A , i s a subset of the r e a l l i n e , i s a measurable function, Suppose  p  is a  and  c:  00  denotes the Euclidean norm.  a - f i n i t e measure on  of underlying p r o b a b i l i t y d i s t r i b u t i o n s .  8 Let  which dominates the family p(" |'6) , 6 e 0 , denote  the density of the p r o b a b i l i t y d i s t r i b u t i o n corresponding to assume  0 x X •> (0, )  0 .  We  p('|") i s j o i n t l y measurable i n i t s arguments.  We r e s t r i c t ourselves to nonrandomized decision r u l e s , which are <f>: X -* A , and define the r i s k of  measurable functions  r ( * , e) = / L O K x )  , e, x) ( x | e ) d ( x ) P  P  <j> to be  .  (1.2.2)  X Suppose A procedure  II i s a p r o b a b i l i t y measure on the Borel subsets of $  i s said to be Bayes with respect to  R(* ) = * J N  F  0 .  II i f  R(<j») < » , (1.2.3)  where  R(<J>) = / r(<fr, 6) dn(8) . 0  4>depends on  II only through the posterior p r o b a b i l i t y d i s t r i b u t i o n ,  PJJ , defined by  p  (B|X = x) = / p(x|6) d n ( 9 )  a.e. [p]  B  for a l l Borel subsets, B , of  0 ,  In f a c t ,  (1.2.4)  / w(e, x) c C e , x) P C d e | x = x) n  <f> (x) =  ,  (1.2.5)  / c C e , x) P ( d e | x = x) e . n  providing i t s Bayes r i s k , R( t'jj) » i s f i n i t e . (  From some points of view i t i s reasonable o - f i n i t e measure.  Providing  Pjj(o[x = x) <  a p r i o r measure (improper i f t e r i o r d i s t r i b u t i o n on  0  00  to allow  II to be a  , a.e. [p] , II i s c a l l e d  H(Q) = °°) . We can define the formal pos-  using  (1.2.4) . A formal Bayes estimator i s  defined as any measurable function on  X , which, evaluated at  x ,  minimizes and makes f i n i t e  / L ( t , 6, x) P ( d 6 | x = x) 0  (1.2.6)  n  a.e.  [p] as a function of  t .  I f such a procedure e x i s t s , i t i s u n i -  que, and given by (1.2.5) , except, possibly, on a set B  for which  / dp(x) / c(0, x) P ( d 0 | x = x) = 0 . B 0  (1.2.7)  n  The condition  SO. 0  + w (0, x)) c(6, x) P^Cdelx = x) < » a.e. [p] 2  i s s u f f i c i e n t to insure that spect to  n  i s a formal Bayes estimator with r e -  II .  A rule that  <j>  (1.2.8)  <j) i s said to be admissible i f there i s no r u l e  rC<j>*, 6) <_ r(<f>, 0)  for a l l 0 e 8 ,  with, s t r i c t i n e q u a l i t y for some  0  q  E 0 .  <|>*  such  (1.2.9) I f II i s a measure on the  7  Borel subsets of with respect to  0 , then a rule IT  <j>  i s said to be almost admissible  i f there i s no rule  <j>* for which (1.2.9) holds,  with s t r i c t inequality on a set of p o s i t i v e  II measure.  C l e a r l y , any  Bayes (but not formal Bayes) rule i s almost admissible with respect to the p r i o r from which i t i s constructed.  In the following sections, we  w i l l be able to apply the following theorem, which gives conditions under which a d m i s s i b i l i t y follows from almost a d m i s s i b i l i t y .  Although  this theorem i s well-known, we w i l l present the short proof for completeness. L('»  I t i s important  to observe that only the s t r i c t convexity of  6, x) , for a l l 8  Theorem 1.2.1; for which  P  and  x , i s used i n the proof.  Suppose for every element (B) > 0 , II({6:  Q  8  q  e ©  and every set  B e B  P (B)}) > 0 .. Then, i f ty i s almost Q  o admissible with respect to  II , i t i s admissible.  m i s s i b i l i t y follows from almost a d m i s s i b i l i t y i f  In p a r t i c u l a r , ad(P J  8 e 0}  Q  is a  8  family of mutually absolutely continuous p r o b a b i l i t y measures. Proof:  Suppose that  <ij  with s t r i c t i n e q u a l i t y at i t follows that  P  Q  Q  (B) > 0 .  n  8  i s not admissible, and that .  If  Let  0  { We have  B = {x e X:  ty*(x)  ^ = ty + ^ ty* . i.  v(ty*, 6) £ r(ty, 8) ,  2.  4 ty(x)} ,  Then, since  < |-L(<(>(x) , 8, x) + -| L(cf)*(x), 8, x)  x e B  = L(<J)(x), 8, x) = L(<f>*(x), 8, x)  x e B  C  (1.2.10)  r ( y , ei < . | r G h e) +1  (<j>*; e) ,  r  (1.2.11)  8  with s t r i c t i n e q u a l i t y i f and only i f  P  QCB)  >  0 .  Therefore,  r(<j>, 6) - r(V, 0) .> 0 , with s t r i c t i n e q u a l i t y i f and only i f  P (B) > 0 , Q  and the conclusion of the theorem follows. We are now i n a p o s i t i o n to state the main r e s u l t of [45], which i s central to this  chapter. II i s absolutely continuous with respect to Lebesgue  Assume that measure on  0  and denote i t s density by  TT .  for almost a d m i s s i b i l i t y involve a function  M:  The s u f f i c i e n t X x 0 ->  conditions <») defined  by 0 -1 M(x,  0) = [c(0, x)p(x|0)Tf(0)]  u  / 0  [w(t, x)-<)> (x)] c(t, x)p(x|t)Tr(t)dt . n  (1.2.12) Also l e t  h(t) = / M (x, t) c ( t , x) p(x|t) dp(x) , X  (1.2.13)  2  and assume Cl) Tr(t) h(t) i s bounded away from of (II)  p(x|e) > 0} i s an i n t e r v a l a.e. [p] .  Under assumptions Cl) and CH) , <f> II  admissible  i f , f o r a l l c e (0„, 0 ) , when  0  0  U  (i)  i s almost  n  with respect to  / c  on compact sub i n t e r v a l s  0 ,  {0;  Theorem 1.2.2;  CD  0  U  TtCt) rOfijT, t ) d t - ~ ,  c / TlCt) r C * , 0, u  t) dt =  -1  Cii) / c  CirCt) hCt))  Cii)'  CTTCC) h(t))  /  X  dt =  00  dt = « .  i  and when  9  In [47], 1.3  [45], c  and  w  did not depend on  x , but as pointed out i n  no d i f f i c u l t y i s encountered i n this case. The Reduced Problem Theorem 1.2.2  can not be applied d i r e c t l y i n our problem because the  mean and variance both are assumed unknown, and therefore the parameter space i s not a subinterval of the r e a l l i n e .  However, i f we r e s t r i c t our-  selves to scale-invariant decision r u l e s , then i t i s possible to obtain a reduced problem which can be formulated as i n Section 2.  In this section  we s h a l l consider a more general problem i n which such a reduction i s possible. As i n [47], we therefore consider a problem which remains invariant under a group group.)  G .  (In our applications, G G  Suppose  i s a transformation group acting on the l e f t of a  given sample space, (X', 5') values i n  X' .  , and that  The d i s t r i b u t i o n of  to be a member of  w i l l represent the scale  {P .*. 6'e 0'}  X'  X'  i s a random v a r i a b l e taking  i s unknown, but i t i s assumed  , a family of d i s t r i b u t i o n s indexed by  Q  o  a parameter set  0'  i s also equipped with a  X' = G/H x X'/G  , where  H  space of l e f t cosets of  H , and  X'/G  Assume that the  0' .  under the equivalence, x^ ~ x^  o-algebra  C  G , G/H  i s a subgroup of  . is  i s the quotient (orbit) space  i f and only i f x^ = gx^  for some  g e G .  (Some situations i n which such a decomposition exists may be found i n Berk 12].)  For s i m p l i c i t y , l e t G* = G/H  Denote the canonical mapping of G  acts on the f i r s t co-ordinate of  and  g e G , then  well-defined byform  G  and onto  X' .  X' = (G*, X*)  0  .  Let  p*  G*  by  .  T * , and assume that  That i s , i f x' = (g*, x*VeX' ,  gx' = (gg*» x*) , where i f gg* = x*(gg } .  X* = X ' / G  g* = x*(g ) , then Q  gg*  is  The random v a r i a b l e , X' , i s then of the  be a  a - f i n i t e measure on a  a-algebra,  10  8* , of subsets of d i s t r i b u t i o n of p* .  X*  X* , and assume that, for each given  0'  6'e 9*, the marginal  i s absolutely continuous with respect to  Denote the corresponding density by  p(x*|0') , and assume i t i s  j o i n t l y measurable i n both arguments. Assume that the action space, A' , i s a subspace of the r e a l l i n e , and that the loss function, L':  A' x 0'  [0, ) , i s of the form 00  L'Ca«, 0') = aC9')||a' - gC9')|| ,  (1.3.1)  2  where  a:  0' -*• (0, °°)  and  g:  0' -»• A'  are measurable functions.  assumption that the problem remains invariant under istence of transformation groups and  A' , respectively.  that i f g  and  g  G  and  G  G  The  e n t a i l s the ex-  acting on the l e f t of  0'  These groups are required to act i n such a way  denote the homomorphic images of  g e G  in  G  and  A  G , respectively, then for a l l on  P_  (gB ) = P ,(B') and 1  fl  g e G , a'e A' , 0 ' e O ' , and  L ' ( g a \ g0') = L ' ( a \ 0')  B'e B' . Assume  H  acts  trivially  A' . A nonrandomized estimator  g S(g*, x*) = S(gg*,x*)for a l l  £:  X' -* A'  g e G  and  i t follows that any invariant estimator of the form If  GT(X*) , where  T:  X* •> A'  i s c a l l e d invariant i f (g*, x*)e X' .  £  If we l e t g=gg  i s equivalent to an estimator  i s a measurable function.  6CG*, X*) = GT(X*) i s an a r b i t r a r y , invariant, nonrandomized es-  timator, then the r i s k of  E , fl  d, r'(6, 0') , i s  aCe')||GT(X*) - B(9')||  2  (1.3.2)  e  _-l _"1 , = E , a(G e')||T(X*) - g(G 0 * > 11  ,  11  because of the assumed invariance of the l o s s . Eg,  Here, and elsewhere,  w i l l mean expectation when the true underlying d i s t r i b u t i o n has  parameter  0 ' . I t follows that  r(fi.e') = E ,(E e  _-l _ - l o Ca(G 0 ' ) | | T ( X * ) - 3(G 8')|r|X*))  Q f  = E , ( c ( e ' , x*)|| T ( X * ) - w(e',x*)|| ) + b ( e ' ) , 2  Q  (1.3.3) where  _-l c(e', X * ) = E , (aCG  0')|X*)  ,  (1.3.4)  D  w(0',  _~1 _ - l X * ) = E ,(a(G 0') B(G 0') |X*)/c(9 ', X * ) , 2.(a(G 0') g (G 0')) fl 1  and  (1.3.5)  e  b(8') = E  1  Z  - E , ( c ( 0 ' , X * ) w (0', X * ) ) .  (1.3.6)  2  e  Let  T  denote the canonical mapping of 0'  onto  0* = 0'/G , and  assume that there exists a (1:1) measurable function ty: Q* -> 0 such that  T0<j>: 0*  0*  i s the i d e n t i t y mapping.  selects one element out of each equivalence  1  In other words, ty  class to represent  that c l a s s .  Wijsman ([41], [42]) has given conditions f o r the existence of such a measurable cross-section.  From the invariance of the problem, i t follows  that  r(6, 0') = r ( 6 , <j>oxC0')) ,  for any invariant r u l e  6 , and for a l l 0'e 0  (1.3.7)  (see, for example, [14],  12  p.  149). I f the choice of decision rules i s r e s t r i c t e d to the class of non-  randomized, invariant procedures,  D^. , a reduced problem i s thus obtained  which can be described as follows. A random variable i s observed. X*  The density (with respect to  i s a unique, but unknown, member of  action space f o r the problem i s L*(a, e*.  A'  X*  p*) of the d i s t r i b u t i o n of  {p(• | <f> (6*)) :  0*e 0*}  .  2  (1.3.8)  + b(<f>(0*)) The class of decision rules a v a i l a b l e , D* X* ->- A '  The  , and loss i s measured by  X*) = c(<j,(e*), X*) ||a-w(<j>(e*) , X * ) | |  functions, T:  X*  with range  , consists of a l l measurable  .  The following obvious r e s u l t s are stated for future reference. A  Lemma 1 . 3 . 1 :  The invariant procedure  loss i s measured by T(X*)  GT(X*)  i s admissible i n  when  L ' ( . , . ) , i f and only i f , for the reduced problem,  i s admissible i n  D*  , when loss i s measured by  2  c(», • ) ||« - w(»,» )||  A  Lemma 1 . 3 . 2 :  The procedure  GT(X*)  i s (formal) Bayes i n  to a p r i o r measure, II' , when loss i s measured by i f , f o r the reduced problem, T(X*) to  n'o T  -1  L ' ( • , • ) , i f and only  i s (formal) Bayes i n  , , . , , , when loss i s measured by  ,  ,  with respect  D*  ,  11  c(«, • ) 11•  , with respect  .112  - w(«, « ) |  In order to apply the r e s u l t s of Section 1.2 and, i n p a r t i c u l a r Theorem 1 . 2 . 2 , we i d e n t i f y  X*  with  that i t i s necessary to assume that  X  and  i s a p r i o r d i s t r i b u t i o n on  t r i b u t i o n on  with  0* = (0*, 0*)  J  II'  0*  V  0' , and  \x  II = II'o T  0* , then we w i l l assume that  0 .  I t follows  , - «> < 9* < 0* < — I — u —  00  .  If  i s the induced d i s -  II has a density  IT with  13  respect to Lebesgue measure. GTJJ(X*)  i s Bayes i n D  /  Then, according to Lemma 1.3.2 and (1.2.5) ,  with respect to  w(8*,  x*)  c(6*, *) x  T (x*) = *  II' i f  TT(9*)  p( *|e*) x  d6*  :  G  /  c(6*, x*)  p(x*|e*) TT(6*)  , (1.3.9) d0*  0* except f o r values of  x*  i n a measurable subset  A C X* > f o r which,  / dp*(x*)7 c(9*, x*) p(x*|e*) ir(6*) d9* = 0 , A 0*  (1.3.10)  and provided that  /  L*CT Cx*) , 9*, *) p(x*|e*) Tr(e*) d0* < » a.e. [ *] . n  x  p  (1.3.11) Here, 6*  represents  <J>(6*) > and we s h a l l continue to follow this  practice when no confusion a r i s e s . Let  M  and  h  be defined as i n Theorem 1.2.2 (with  x* , 6* , 9* , 9* , p* X, u for any  replacing  x, 9, 0 , 6 ~ u  p,  r e s p e c t i v e l y ) , and  T e D* , l e t  r*(T, 6*) = / L*(T(x*), 0*, x*) p(x*|9*) dp*(x*) . X*  Theorem 1.3.1: Under assumptions (I) and (II) , GT^(X*) m i s s i b l e with respect to  n'  (1.3.12)  i s almost ad-  i n Dj. i f , f o r a l l c e (6*, 6*)  , when  14 6 (i)  * u Tr(t)  /  r*(T  t)  ,  dt  =  » ,  then  n  c 0  (ii) / Gr(t) h ( t ) ) " c U  and when ( i ) '  /  dt = oo ,  1  ir(t) r*(T , t) dt = °° , then  C  (ii)' /  (ir(t) h(t))  -1 1  dt = co .  In the following sections this theorem w i l l be applied i n two  special  cases i n v o l v i n g the normal law.  1.4 Quantile Estimation In t h i s section.we consider the problem i n which we observe dent random variables  X  and  Z=  2  and  Z:  ^ ^»  °" I) •  n  l  n  (Z,, Z ± z 0  Z )' , where n  indepen-  X: N ( , u  2 a  r  the usual s i t u a t i o n , i n which we have inde-  pendent observations from a normal population, we obtain this model 2 after applying a s u i t a b l e orthogonal transformation. are assumed to be unknown and we wish to estimate  Here, a  and  u + ncr > where  p n  L(a; a) =constant, a ||a-u-noj| A s u ffunction f i c i e n t sitsa tof i s tthe i c iform n this proble i s a u, known and our• loss 2  2  ' n  is  (X, S) , where  S =  lem  2  z Z. i=l  » and we need consider only estimators  1  that are a function of this s t a t i s t i c .  The problem remains i n v a r i a n t  under the transformation group G^ such that (X, S) (cX + d, c S) (li, cr) (cu + d, ccr) 2  (1.4.1)  S  )  15  where  c > 0  and  *-<» < d < «>.  It follows that any nonrandomized  Gj-  1/2 invariant estimator of Since  G-^  y + na  X + cS ' ,-«> < c < » .  acts t r a n s i t i v e l y on the parameter space, there exists a  best choice of  c .  However, Zidek [46] has shown that the r e s u l t i n g  estimator i s inadmissible i f the subgroup of exists a  i s of the form  G^  n. ^ 0 .  In f a c t , i f we l e t G^  obtained by putting  G2 invariant -  d = 0  denote  i n (1.4.1), then there  ( i . e . scale-invariant) procedure having uniformly  smaller r i s k . In this example, i t i s easy to see that the results of the previous G= G = G = G2 > ti ={e}  section apply with X' = ( G * , X*) = ( S  1 / 2  , XS~  6* = (y, a) , x(e') = y a "  7  1  notation we w i l l denote X*t Let I (a, b) = tV n  1/2  , and  <j>(e*) = (6*, 1) .  For s i m p l i c i t y of  Y and 0* by X • ~ C " > / dt .  2 / 2  G  X* = 0* = ( - c o , 00) , G* = (0, c o ) ,  ) ,  by  = the i d e n t i t y element of  at  b  2  (1.4.2)  2  0  Then, i f  p*  i s Lebesgue measure, i t i s e a s i l y shown that  p(y|x) = (constant) I ( y , X)  (1.4.3)  n  cCX, y) = E C s | Y = y ) = I x  and  w(X,  n+2  ( y , X ) / I ( y , X)  (1.4.4)  n  y) = (X + n) E,CS |Y=y)/c(X, y) 1/2  = (X + n) i  (y,. >/ x  n + 1  I  (y» > A  n + 2  •  F i n a l l y , the formal Bayes estimator with respect to  1/2 exists) i s  S '  T (Y) n  , where  (1.4.5)  n (assuming one  16 OO  / (X + n) I  +  1  Cy,. x)  TT(A)  dx  —CO  (1.4.6)  00  . ( y , x) TT(X) dx  /I  9  —CO  In order to obtain an applicable consequence of Theorem 1.3.1, assume al.  TT(X)  a2.  ir i s non-increasing on on  >  0  ,  X < co  -°° <  TT i s non-decreasing  (0, °°) , and  (-«>, 0) ,  a3.  TT i s bounded,  a4.  ir(sX)n  —1 where  Theorem 1.4.1:  t  —8 (X) _< c  c. > 0  1  s  and  , for a l l  9  X  and  0 < s < 1 ,  g < n + 2 .  Under assumptions a l , a2, a3, and a4, S  '*" s  admissible within the class of scale-invariant procedures i f (i)  T n  ( y ) -y  i s bounded, CO  Note:  Explicit  conditions on  TT f o r the boundedness of  T (y) -y  w i l l be presented i n Theorem 1.4.2 . Proof:  The r e s u l t w i l l follow from Theorem 1.3.1 i f we can show that  a l , a2, a3, a4, and ( i ) together imply  h(X) < KX  2  |X| > 1 .  Note that here, and throughout the thesis, K  (1.4.7)  w i l l be used to denote a  17  generic constant whose precise value i s of no relevance to the argument. Recalling  equation  M(y,x) i  n  +  (1.2.12) we see that  (y, X ) TT(X) = / t i  2  n + 1  (y,  t)  ,r(t)dt  X  + n / I  ( y . > ir(t) dt - T ( y ) / I fc  n + 1  n  X  n + 2  ( y , t)ir(t)dt  X  But, a f t e r interchanging i n t e g r a l s , the f i r s t term i s equal to  °° n+1 -x /2 " -(xy - t ) / 2 / x e ft e Tr(t)dtdx o X 2  2  w  °° n+1 -x /2 °°. -{xy - t ) / 2 , . , , / x e / ( t - xy)e ^ ir(t)dtdx o X 2  2  s  a  °! n+2 -x /2 °! -(xy - t ) / 2 , . , . + y/ x e / e ir(t)dtdx 2  2  w  It follows that  M(y, A ) i  n + 2  (y,  x)7r(x) = (y - ^ ( y ) ) / i  n + 2  ( y , OuCOdt  X  + n / i  n+1  ( y » t)u(t)dt  (1.4.8)  X  , +  2 2 °! n+1 -x /2 °! . -(xy - t) /2 ^ v , , , / x e ' / ( t - xy)e ir(t)dtdx o X  oo  00  s  w  18  Therefore, using a2 and ( i ) , f o r X > 0  M (y» ) 2  X  +  I  2 y' (  n +  n+l  L  X  "  C A y  i  )  1 )  J  +  K l H  ( s 8 n  nf2  y  ( y  '  X  )  +  H  n+l ' ( y  X  )  +  ^ l ^ '  X  )  (1.4.9)  )  where, i f we l e t  2 f(x) = ( 2 n ) "  then  1 / 2  e"  X1  1  ,  (1.4.10)  H ( a , b) - / I ( a , t) dt n  n  b  and  L (a) = / f ( x ) d x , a  (1.4.12)  n  x  Here, J ( s g n y) i s equal to 1 i f y +  i s p o s i t i v e , and i s equal to 0  otherwise. The l a s t term i n (1.4,9) a r i s e s , since f o r y > 0  v-\x)ff  x  e~  n + 1  o  oo  , n+1 <. / x o  e  x 2 / 2  Jct-xy) e ~ X  C x y  -  t ) 2 / 2  *(t)dtdx|  2 . oo , 9 ~x 2 . | , -(xy-t) /2 , . / I t-xy| e ' ' dtdx X w  and X > 0 ,  19  2 2 . n+1 -x /2 . I , -u 12 . . /x e / IuI e dudx o X-xy 00  oo  \  n+1  X  f x  , 7 +/^x Xy  If  ( y  y < 0 , then  remove  r,  X )  ~ / X-xy  -u /2 2  ue  7 i i / l X-xy  ~ /2 e  n + l  - n+l » I  -x /2 2  e  x2  ^rri-l^ "^  +  7  t - xy > 0  dudx A  - / « , ,, | dudx u 2  u  2  e  *  for a l l ^ +±(y'  d i r e c t l y , obtaining  A  n  (1.4.13)  t > X > 0 , and therefore we ^  a s  t n e  can  s q u i r e d upper bound.  Before continuing with the proof of the theorem, i t w i l l be useful to have a v a i l a b l e the following lemmas.  Lemma 1.4.1: (a)  If  (-»  < n < °°)  K. < l T ' 1 1 /  a > a — o  2  2  < K„ , then there e x i s t s Z  n / / 2 _ 1  a  o  implies  L (a) n  K  <_  < K 2 e"  (b)  There e x i s t  a  / 2  (l+a /2) 2  K„, K. > 0  ( n  -  1 ) / 2  such that  a >_ 0  implies  such that  L (a) n  K  3  - 4 K  e-  Proof:  a 2 / 2  Cl  a /2) 2  +  ( n  -  1 ) / 2  By d i f f e r e n t i a t i n g both of i t s sides, the following i d e n t i t y  proved:  e-  a 2 / 2  ( l + a /2) 2  C n  "  "  =  1 ) / 2  e-y[-2-\n-l)  (l y)  ( n  +  ~  3 ) / 2  a /2 2  + d+y)  ( n  -  1 ) / 2  ]dy .  2 Thus, e ~  a / 2  (1 + a /2) 2  /  y  C N  -  1 ) / 2  ( n  ~  may be written as  1 ) / 2  I-2- (n-3)y1  C n  -  (l y)  1 ) / 2  ( n  +  -  3 ) / 2  a /2 2  +  y-  ( n  -  3 ) / 2  (l y) +  ( n  - ) 3  / 2  ]dy  To prove (a), note that  ? -y (n-l)/2 . l-n/2 / e y dy = 2 a /2 0  J  _l/2 _ . . L (a) , a >^ 0  II  2  Thus, since  J £  [  _ -l 2  ( n  .3  ) y  -(n-l)/2  a 4 7 )  (n-3)/2  +  y  -(n-3)/2  ( 1 + y )  (n- y 3  2 ]  =  ±  21  the l i m i t of the r a t i o i n Ca) i s 2 ° ^ (b)  II  2  2  .  2  follows from Ca) and the continuity of L ^ a ) e  a / 2  C1 + a / 2 ) 2  ( 1 _ n ) / 2  For convenience i n the sequel, we set  t  / f(x)dx- C2lir  FCt) =  2  t 1 / 2  / e~  X1  dx ,  1  J (a) = / x . f (x-a)dx , n o  (1.4.15)  n  and  i  R (a) = J (a) J * (a) . n n n+1 -  Lemma 1.4.2:  R (a) £ a "  (b)  There exists  a >0  1  n  K,. such that  R (a) < K_ n — 5 (c)  a >0 . —  I f a < 0 , there exists o R (a) < K. l a l n — 6 ' There exist  and Kg  R (a) < K., lal + K n — 7 ' 8 1  6  such that  a < a . — o  1  (d)  (1.4.16)  (n = 0 , 1, 2, ... )  (a)  Q  (1.4.14)  such that  a < 0 .  22  Proof:  Since  x  e q u a l i t y that  i s convex f o r x > 0 , i t follows from Jensen's i n -  R (a) > R n  (a) , Therefore, R (a) < R (a) . But n  n+1  R (a) = F(a)IfCa) + aFCa)]" o  < a" —  1  o  a > 0 .  1  The remaining parts of the lemma follow from the continuity of from the observation that, by l'Hopital's r u l e ,  lim F ( a ) [ f ( a ) + a F ( a ) J a-*—°°  |a |  _ 1  = l i m [2|a| F ( a ) f ~ ( a ) - 1 ] ~ a-*— 1  1  = 1  1  00  (For a proof of the l a t t e r equality refer to [12], p.166.)  Lemma 1.4.3: (a)  There e x i s t  K  (b)  (n = 0 , 1, 2, ...)  y Q  a  n  There exists  If ' a  such that  < J (a) < BT (a + l ) - n - 10  J (a) > K,, n — 11  (c)  K ,K > 0 y 10  *  0  > 0  a > 0 .  n  such that  a > 0 —  there exist  K  1 2 >  K  1 3  > 0  such that  R^  and/or  23  ,  K T  I  a 12 ' 1 0  I  2 -a /2 T, i l n-1 e < J (a) < K » a — n — 13  -n-1  /  T  1  \  2 -a /2 e  a < a — o  .  Proof: (a)  From Lemma 1.4.2  (a)  J (a) > aJ .(a) > ... > a°J (a) = a F(a) > 2 n — n-1 — — o —  —  Also, J (a) = n  < 2  n  1  (a  1 1  +  / (x+a)  n  n  1  f(x)dx < 2 ~ (a  F(a) +  n  l  a  a > 0 . —  U  w n  /|x| 1  1  / |x|  n  f(x)dx)  a > 0 .  (b)  This r e s u l t i s t r i v i a l .  n  (c)  Since  J (a) = Z n .  H  lija  1=0  i t follows that, for  i  L .(-a) , n-i  a < a  , o  i=o  1=0  f(x)dx)  -a  The r e s u l t follows from continuity and the fact that for  n  (a+l)  n  > 0  24  where  =  max K ^ ( i ) . Here, f o r -co < o<l<n  accent the dependence on n  Therefore, J (a) < K  n = K2 v o  < » , K^(n) i s used to  n  I 1  a 1  i n Lemma 1.4.1 .  Z (i)|a| |a| i=o 1  n - i - 1  &~ ' a  ?/2  2 |n-1 -a /2 e  a < a . — o  Also, by Lemma 1.4.2(c) ,  J ( a ) ^ K| a |  -  1  n  J  x  ( a ) >. . .. >. K| a |  J ( a , 0)  _ n  = .K,|.aj F(a)  a <a ~ o  n  Therefore, since  J (a) > Rial n — 1  Lemma 1.4.4:  1  where  H n  1  n  1  f(a)  If X > 1  H (y, A) < K n  l i m |a| F ( a ) f ( a ) = 1 , a->~ oo  e-  a <a — o  and y > 0 , then  x 2 / 2 ( 1 + y 2 )  (l X /2(l y )) 2  +  i s as defined i n (1.4.11)  2  +  ( n  -  1 ) / 2  ,  25  Proof:  Assume throughout  I (y,  X) - ( 2 n )  n  the proof that  (l yV  1 / 2  ( n + 1  +  >  / 2 e  X > 1  -*  2 / 2 ( 1 + y 2  and y > 0 .  >  J (yX(l yV n  1 / 2  +  ) ,  (1.4.17) and therefore, by Lemma 1.4.3,  H (y, X) < ( 2 n )  1 / 2  n  (l y )- " 2  (  + 1 ) / 2  +  K  l p  7  . - ' ' " ( ^ ' ( y t d V ) "  '  1  2  +  1>"  X  < (2n)  1 / 2  (l y )2  ( n + 1 ) / 2  +  K  1 e ~ X  1 Q  t  2  '  2  ^ ( t l ) dt n  +  X = K L (X(l+y )" 2  1 / 2  n  < KK  4  e-  A 2 / 2 ( 1 + y 2 )  )  (l X /2(l y ))( - )/ 2  +  2  n  1  +  2  ,  by Lemma 1.4.1.  We now return to the proof of Theorem 1.4.2.  Assume, f o r the present,  °° 2 that  X > 1 , and r e c a l l that  h(X) =  / M (y, ^ )  I n + 2  (y»  *)dy •  —OO  the proof i n this case w i l l be complete i f we can show that  (I)  1  / —00 0  (II)  /  2 M (y, X ) I  n + 2  ( y , X)dy < K ,  n + 2  ( y , A)dy < K ,  2 lC(y,  X)I  T  h  u  s  26  (III)  2 / lT(y, A)I  (IV)  / M (y, X ) I  (V)  / M (y, A ) I A  1  and  (I)  n+2  ( y , . A)dy < K ,  n + 2  ( y , A)dy < KX ,  n + 2  ( y , A)dy < KX  2  2  2  2  (-°° < y < -1);  As a consequence of (1.4.17)  |H  n+2  (y, * ) / I  ~S e -  n + 2  t 2 / 2 ( 1  (y. »  l= e  ^ >  (ytU  2  J n + 2  x 2 / 2 ( l  ^ >J^ (yA(14y )2  2  1 / 2  2  2 + y  )-  1 / 2  )  )dt  A < KK  13  K"  Similarly, | H  Also, | i  n + 1  X  1  n + 1  2  1 / 2  < K (l+y ) 2  e  R  1 / 2  ( y t t f 2  L ^ A )  X 2 / 2  (y, A)/I  (y, ^)/i  = (l+y )  6  n + 3  n + 1  '-  (y,  A)|±KX  x )  (yA(l+y ) 2  n + 1  |y|(l+y )2  < KK^ X  1 / 2  1 / 2  )  A  2 n + 1  n + 3  (l+X /2) 2  n / 2  < KA  2 n + 3  < KX(l+y ) 2  .  1 / 2  Moreover, |H (y,. A ) / I n+1  = |H  < KX  n+1  2 n + 2  Therefore,  -  (y, X)/I  (l  )  2 + y  n + 1  n + 2  ( y , X)|  (y,  X)|.| I  n + 1  ( y , X)/I . (y, X) | jri  2  •  1 / 2  for a l l X > 1 ,  2  1  / M (y, X ) I  n + 2  ( y , X)dy  —CO  < K "} X  M  a  +  y  2  ) - ^  +  1  )  /  e-^^^^J^CyXd+y )2  2  1 7 2  —CO  < KK  X  13  e"  5 n + 5  ') ( l y ) -  x 2 / 2  2  ( n + 1 ) / 2  +  dy < K .  —CO  (II)  (-1 < y < 0):  Since  ^ +2^"^ ^ i s  ncreas:  n  |  /  U  ,w  *- S> n  r  T  , M  X /2(l+y ) " 2  2  A - (l+y ) 2  1 / 2  f" (X(l+y )(l-F(X(l+y )~ 1  < X (l+y ) < 2X _ 1  2  2  _1  .  2  1 / 2  )  -t /2(l+y ) 2  2  J t  28  (The second l a s t inequality i s given i n [12], p. 166)  Also, | i  n + 1  ( y , *>/i <y.. A ) | n+2  = (l+y ) 2  1 / 2  R  (yX(l+y )" 2  n + 1  1 / 2  )  <_ K(K A + Kg) . ?  Morover, proceeding as i n ( I ) ,  |H  (y, A)/I  n+1  Therefore, since  °f M (y, X ) I  J .,.(•) i s increasing,  2  (Ill)  ( y , X)| < K^+KgX" ) . 1  n + 2  ( y , X) < K ( K + K X ) e " _1  n + 2  7  A  < K .  74  8  (0 < y < 1) :  |H  n+2  (y, X)/I  Similarly, |H  n+1  n + 2  ( y , X)| < K K  (y, X)/I  n + 2  K ~ (l+X /2(1+y )) 1  14  2  ]L  ( y , X)| < KX  n  .  2 1/2  Also, | l  n + 1  ( y , X)/I  (xy  - 1  )/l  n + 2  + 2  ( y , A)| < K (l+y )  ( y , A) |  5  < K ,  2  (  n  +  1  )  /  2  < KX  n + 1  .  < |L  (x(l y )2  n + 1  < (2n)-  < KX  )/i  K,(i y )(  1 / 2  2  n + 2  n + 3  +  >  ( y , X)  / 2  (i x /2.(i 2  +  2 + y  ))  n / 2  ^ (yAa+yV 2  1 / 2  )  n  Therefore, f o r a l l  1  1 / 2  +  X > 1,  2  / M (y, X ) I o < x  2 n + 2  ) i (  n + 2  ( y , X)dy  2 + y  )-<"  + 3  >'  e-  2  i 2 / 2 ( 1 +  "  2 )  V 2  ( x(i y )y  +  2  1 / 2  )ay  2n+2 ^ .n+2 -X /4 < K . (X+l) e — 2  —  10  /1  1  (IV) (1 < y < X)  Here, l e t f ( y , X) be any function f o r which  |f(y, X)| <. K X ( l + y ) ^ m  Then, from (1.4.17), i t follows that  X  2  / f (y, X)I Z  <KX  n + 2 + 2 m  /  n + 2  ( y , X)dy  (l y )^ 2  +  2 k  - n  3 ) / 2  e-  x 2 / 2 ( 1 + y 2 )  dy  2  k  2  2^1/2 f ' 2k-n-3 , 2 - l / 2 J u [(u -1) l/2  /•T J.T.  ,n+2+2m = KX  K  Tr  {  . -X u] e  l N  2  < KX  J l/2  n + 2 + 2 m  u  2  k  —  e"  3  A /  2  U  e"  V 1  du  2  - KX  X2 J  2 ( m + k )  v  x(i x )2  +  < KX  n + 1  "  2 k  1  dv  1 / 2  2  providing that For  2  _ 1 / 2  m + k •<_ 1  y > 1 ,  Therefore, |H^. (y,. ) / x  I  2  < KK  y' ^ X  X- - (l X /2(l y )) n  U  ( n + 2  2  2  2  +  +  < KX ( l + y ) , 1 < y < X 1  Similarly,  2  J H ^ C y , A)/I  n+2  <y» A )  ( n + 1 ) / 2  (l y ) 2  +  ( n + 3 ) / 2  2  <KX- - (l X /2(l y )) n  2  2  2  +  < KX  -2  n+1  < l\  <  (l y ) 2  ( n + 3 ) / 2  +  2 3/2 (1+yT  |L (Xy"Vi (y,  Also,  n / 2  +  n+2  ( (i y )~ x  + 1  +  2  1 / 2  x)|  )/i  (y,  n + 2  KK X " (l+X /2(l+y )) _ n  2  2  2  n / 2  4  < KX- (l y ) 2  2  x)  (l+y ) 2  ( n + 3 ) / 2  3 / 2  +  Moreover, J ' I  - d+y ) 2  (y, ) x  n+1  1 / 2  |R  < X" (l+y )y" 1  2  1  / I  '(y' > I x  n + 2  (yx(i+y )"  1/2  KX~ (l+y )  1 / 2  2  n+1  <  1  2  )  In each case the bound i s a function of the form m + k < 1 .  (V) (X < y < ») Using (1.4.6)i we see that  KA^l+y )^ 2  2  , where  32  / C n T  ( y ) I n  +2  ( y  ' O-Ct+^I^Cy,  and i t follows from (1.4.8) that  M  (y>  M  t)]ir(t)dt = 0 ,  (1.4.18)  also s a t i s f i e s  X)I . ( y , A)TT(A) = (T (y)-y) / I ..(y, t ) i r ( t ) d t 9  n  -n  / i  7 -fx  n+1  n + 1  C y , t)7r(t)dt  -x /2 e  (1.4.19)  \. .. ~ ( x y - t ) / 2 , . /(t-xy)e ' Tr(t)dt dx .  2  2  N J  w  Therefore, M2  2, , 2 , ( y > . > n+2 ' A  w I  ,  (y  %  2,,  v  . , \ < KtC / I r  , , . 2 < y » t)ir(t)dt) %  n+2  v  v  2  X  + .( / ^ ( y . t ) T r ( t ) d t ) '  . T. n+1  +  (/ x  -x /2 2  e  \ .  (1.4.20)  . -(xy-t) /2 2  t  / (t-xy)e  w  ,  X J  As i n part IV, i f f ( y , A) i s any function for which then  / f ( y , X)I X 2  n + 2  „.2(m+k) ^ < KA A  ( y , A)  1 + a 2  <  < KA , 2  f  )  +l-2k v  n  -v /2 , e dv 2  ,2,  Tr(t)dt dx) ] J  |f(y, A) | <. K A ( l + y ) ' m  2  k/  2  ,  providing that see that f o r  /  m + k <_ 1 , and  k < (n+2)/2 .  using  t)Tr(t)dt  - K(l y )-< 2  n + 3  +  < K(l y )-< 2  n + 3  +  < K(l y )2  )  ( n + 3 ) / 2  - K(l y )-< 2  n + 3  +  } e-  / 2  )/  +  [K  2  2  / ( 2  / e1  +  [,(X)X  >/  t 2  y )j 2  t 2 / 2 ( 1 + y 2 )  ,(X)X  e +  r(A)[X X e  T  1 +  g  <K(l y )2  ( n + 3  +  )  / 2  X  N  +  3  (y(l y )2  n + 2  t  (t)dt]  n + 2 1 T  2  2  e  +  t  2  2  +  applied twice. 1  .  < K(i+  2 y  )-  ( n + 2 ) / 2  x  n + 2  TT(X)  F i r s t observe that  ,\ " n+1 \S I x 1 o  -x /2, . -(xy-t) /2 . , .. • (xy-t)e dx ir(t)dt| 2  e  +  ,  u ]  TT(X)  , and i n the second case, s = tX  (y, tMOdt  u  d t ]  1  Similarly,  n + 1  +  -X u /2(l y ) n 2-B 2  e  In the second i n e q u a l i t y , assumption a4 was 1  t)rr(t)dt  1 / 2  +  / -t /2(l y ) n 2-g  )  n + 3  +  X"  / i  (1.4.17), we  y > X ,  ^ ( y .  case, s = X  Now,  2  J  In the f i r s t  ) 7 x 1 0  n  K ( i  +  +  e"  1  V  y  M  ,  2  J " « ' 1 0  t  -  (  x  "  y  )  2  /  dx r r ( t ) d t  2  |wyd+y )-  l  2  t  2  2  1  /  1/2  -t|  -t) /2 2  2  dw i r ( t ) d t  e  W r ^  2  "  2  ) " " ^ 1  -(wy(l+y )" 2  ,  ,  e  -v /2(l+y >-(wyU+y ) 2  ,  |xy-t|  x 2 / 2  1  l»y(i+y )2  2  ( n + 2 ) / 2  +  1 / 2  -t) /2 2  / / (v+t) 1 -t  K(l y )-  ( n + 2 ) / 2  / 7 (|v| 1 -co  K(l y )-  ( n + 2 ) / 2  ir<A)A / 1  2  +  -t|  0  e  K(l y )-  1/2  dw 7r(t)dt  |v| e "  n + 1  + t  n+1  n + 1  V  1  2  dv  ) | v | e"  V  T r  1 2  (t)dt  dv , ( t ) d t  2 2  +  Ka+y )-^ 2  7 2  -^^  3  2  +  < n + 2 ) / 2  X  n + 2  „(X>  n + 1  -00  ) t 1  K(l y )-  7 (|v|  B  n  +  1  '  P  dt  +  t  n + 1  ) | v | e"  V  /  2  dv t "  B  35  F i n a l l y , by treating the cases  -» < t < 0 , and  0 < t < 1  i t i s not hard to see that 1 oo 2 2 , n+1 -x 12. . -(xy-t) /2 , . , [ / / x e (xy-t)e dx Tr(t)dt| -co o N  w  <K(l+y )-< 2  <K(l+y )2  But, f o r y >  I  n + 2  (Y,  n + 2  ( n + 2  X  >  / 2  )/  *  2 X  (X)  v  ,  X) > K ( l + y ) - < 2  > K(l+y )-  ( n + 3 ) / 2  > K(l+y )-  ( n + 3 ) / 2  2  2  X  X  N  +  N  2  +  n+3  E  >/  -  A  2  x  2  /  2  (  e - ^  n + 2  1  +  A  2  >  2  Therefore  /  I  ( y t t f 2  '  t ) i r ( t ) d t  < KX  -CO  i  n + 2  (y  ..  ,  X)TT(X)  / i (y, tMt)dt _ i ( y , X)TT(X)  .  n + 1  n  n + 2  +  1  j  < K(l+y ) 2  1 / 2  ,  /  2  ^  separately,  36  , and  X °°  r , , n+1 j / / x  -x  e  -°° o I (y,  2  12.  . - ( x y - t ) ( x y - t ) e  2  /2  , iT(t)dx S  w  j  , . dt |  __ A)TT(X)  n + 2  < K(l+y ) 2  1 / 2  In each case the bound i s a function of the form k + m <^ 1 , and  Now,  M(y,  k < (n+2)/2 .  )  I n +  -n  ,  +  2  /  "  /  x  ( y  '  x n + 1  A ) 7 T ( A )  (  T  (  y  )  _  y  )  /  n  (y»  n+1  2  M  " /  /-j.  k  2  , with  X > 1 .  also s a t i s f i e s  ^+2^' —A  - t ^ ^ d t  -t)ir(-t)dt  -x /2 2  e  o  Also  =  m  The proof i s complete f o r  using (1.4.19), we see that  x  KX (l+y ) ^  (1.4.21)  \ -(xy+t) /2 2  (t+xy)e  J  ,  . _ At  ir(-t)dt  ,  dx  -X  I ( y , t) = I ( - y , -t) . n  But i f  n  X < -1 , Tr(-t) <. TT(X) for  t > -X .  So we r e a d i l y obtain the  basic i n e q u a l i t y corresponding to (1.4.9), and i t d i f f e r s only i n as much as  -y  and  |x|  replace  y  and  X , respectively.  the previous analysis allows us to conclude that  / M (y, A 2  A)I  n + 2  ( y , X)dy < KX  2  , X < -1 .  Therefore,  37  S i m i l a r l y , using  M  (y»  x  )  I  n + 2  +n  (  (1.4.8)  y»  I  X ) 7 T ( X )  x n + 1  =  (y- (y)> / T  n  1  2-> (  T&  y  -OuC-Odt  ( y . -t)ir(-t)dt  (1.4.22)  — CO  7 -fx  n+1  -x /2 ~ 2  e  X  f  , , . -(xy+t) /2 , . , (t+xy)e Tr(-t)dt dx 2  3  The analysis of part V now allows us to conclude that  / M (y, 2  In  A)I  n + 2  ( y , A)dA < KA  2  , A < -1 .  this case assumption a4 i s applied with The proof of Theorem 1.4.1  s = -A ^  and  s = -tA  i s complete.  Assume: a5.  -1 ir(t)TT~ (A) i .  c  where  Theorem 1.4.2:  + 2  °3  l | t_x  f o r  a 1 1  x  a n d  t  »  c^, c^, a > 0 .  Under assumptions  that  |T (y)-y| < K , -co < y < n  a  a2, a4, and a5, there e x i s t s  K  such  38  Proof:  I t can be shown that  T ( y ) = y + A + nB ,  (1.4.23)  n  " n+1 -x /2 fx e  " \ -(xy-X) /2 . , f (X-xy)e Tr(X)dX dx  2  where  A = ^ . fx  e  -x 12  \2 . , -(xy-X) 2 , f e • 7r(X)dX dx  —  CO , fx  n+2  (1.4.25)  O O  /  -x 12  e  J  . -(xy-X) /2 ,. . , / e ' Tr(X)dX dx SA  2, l -x 2  2 , -u/2  00  00  t|/ x  n  1  +  e  , / ue  , j I iT(u+xy)du dx| /  L  Al , n+2 -x /2 / x e 2  oo / x ^ e 1 1  X  2 ^  2  . -u /2 , . , j f e TT (u+xy) du dx 2  co / |u| e  fr(xy)  U  2 ^  2  Tr(u+xy)Tr "*"(xy)du dx  0  <  n+2 / x  -x /2 e  <K*f(y) ,  2  , • -u /2 it(xy) / e  , , \ -1/ • > , 1 Tr(u+xy)Tr (xy)du dx  2  s  .  2  / |u| e~  U  / 2  (c +c |u| )du a  2  3  —CO  where  ¥> =  !  /e  (1.4.24)  J  00  B = ^  Now,  2 . -(xy-X) /2 /-.x,, , f e Tr(X)dX dx  ~  2  n+2  oo 2 , n+1 -x /2 /x e and  2  J  (c +c |u| ) 2  3  du  39 2 . n+1 -x 12 , . -1. / x e ir(xy)Tr (y)dx  oo  and  f(y) -  Similarly,  o  °° 2 . n+2 -x 2 , > - L >, / x e IT (xy) IT (y)dx  | B | <_ K**f (y) ,  J where  e-  u 2 / 2  (c +c |u| )du a  2  3  K** J e ^ ^ + C g l u l  / rx  n  +  1  v  c  But  f(y) <  ~e / x 2  Q  2  0  1  ) -  1  du  j x. +7 / n+1 -ex .dx x3 d x 2  5  »  3  n+2 -x 12 f x e. 7  and the r e s u l t follows.  1.5  Scale Estimation The structure i n this section i s i d e n t i c a l to that i n Section 1.4, ex-  cept that the problem of i n t e r e s t i s the estimation of a / loss function i s given by L(a; u, a) = a on  A=(0, oo) i s such that  estimator i s of the form  a -> c  2 p  2 p  , p > 0 . The  2 2 |a-a > and the action of  a , c > 0 . Any 6^-invariant  c S , and, again, the best P  G^-invariant estimator  i s inadmissible ([37], [5]). In this example,  c(X, y) = E (S |Y=y) = I 2p  x  n + 4 p  ( y , A ) / I ( y , X) n  (1.5.1)  40  and  w(X, y) = E (S |y=y)/c(X, y) P  X  W  =  '  7  X ) / I  n 4 +  ( y P  '  X )  '  (1  The formal Bayes estimator with respect to  n  - ' > 5  2  (assuming one exists) i s  S T ( Y ) , where, P  n  CO  / • T  n  (  y  )  =  Z  I n+2p  -(y» x)Tr(x)dx  E  •  <--> 1  5  3  Theorem 1.4.1 and Theorem 1.4.2 have analogues i n this example, and the methods of proof are s i m i l a r to, and s l i g h t l y those of Section 1.4.  M(y,. ) i x  n + 4 p  - T (y) / I X n  3 < n+2  Replacing  Theorem 1.5.1:  n + 4 p  by  n + 2 p  (y,  ( y , t)Ti(t)dt  g < n+2p+l  in  Under assumptions  t)Tr(t)dt  .  (1.5.4)  a4 , we have the following theorems:  a l , a2, a3, and a4, S T^(Y) P  within the class of scale-invariant procedures i f  (i)  ^rj(y)  i  s  bounded,  CO  and  than,  We therefore present only the r e s u l t s , noting that  A)TT.(A) = / i X  (y,  less complicated  ( i i ) / (X T T U ) ) "  1  -1 dX = / ( X T r ( X ) ) 2  _1  dX =  i s admissible  Theorem 1.5.2:  Under assumptions a2, a4, and a5, there exists  that  |T (y) | <. K , n  -» < y <  00  •  K  such  42  Chapter 2: Minimax, Scale-admissible Estimators Of Scale Parameters  2.1  Introduction Consider the canonical form of the general l i n e a r model i n which we  observe independent random variables 2 X: N ( u , a I) k  X  and Z = (Z^,  2 2 and Z: N (0, o I) . Here, a ^  unknown and we wish to estimate the present) i s of the form  n  are assumed to be  a (p>0) , where our loss function (for 2p  L(a; u» a) = a  s t a t i s t i c i n this problem i s  and u  Z ) ' , where  (X, S) , where  (a-a ) S=  n Z  . A sufficient 2  2.  , and we need  i=l consider only estimators that are a function of this s t a t i s t i c . problem remains invariant under the transformation group (X, S)  _  (caX+d, c S )  (u, a)  *-  (cau+d, ca)  *-  2p c a  a  The  G , such that  2  (2.1.1)  v  1^  where  c > 0 ,d E R  , and a  that any nonrandomized Since  G  is a k x k  orthogonal matrix.  G-invariant estimator of a ^  P  It follows  i s of the form c S  acts t r a n s i t i v e l y on the parameter space, there exists a best  choice of c , given.by  (s )  E  c°= J ^ o,i(s 1  P  =  9  E  where  r( +f)  p  w  2 p  — 2 r(2p+|)  )  P  1  , x-1 -u , T(x) = / u e du . f \  ,  (2.1.2)  P  43  Although  c°S  i s minimax ([22], [23]), i t i s well-known that this  P  estimate i s inadmissible ([37], [5]). Q  the subgroup of exists an  In f a c t , i f we l e t  obtained by putting  d = 0  i n (2.1.1),  ff denote then there  H - i a r i a n t procedure having uniformly smaller r i s k .  However,  n v  the estimators which have been chosen to dominate the usual procedure are themselves  H inadmissible.  In this chapter, we construct an a l t e r n a t i v e ,  -  minimax estimator, which i s formal Bayes within the class of s c a l e - i n v a r i a n t procedures.  If  k=l., then using the r e s u l t s of Chapter 1, we are able to  show that this estimator i s also scale-admissible.  An extension of these  r e s u l t s to the problem obtained by introducing a more general loss function i s c a r r i e d out i n Section 4.  As a f i r s t step i n the construction, we  will  rederive the estimators of Stein and Brown i n a manner which w i l l suggest the d i r e c t i o n i n which to proceed.  2.2  I n a d m i s s i b i l i t y of the best  Y =  Let  S~  X  1/2  , W  =  ||Y||  2  G-invariant  =  Y  z  2  , = A  estimator  _ a  1  y  6 =  , and  ||A||  2  .  i=l Observe that for any nonrandomized, /-/-invariant procedure, >'(W)S , we P  r  have  E  [o  4 p  (y(W)S P  2 p 0  ) ] = E 2  u>0  . [(Y(W)S -1) ] . (6 fAI^ ,0,...,0),1 P  Our goal i n this section i s that of finding a  2  (2.2.1)  B -measurable function, Y , w  such that  sup E rGKW)S -l) ] < E [(c°S -l) ] , 6>0. ° p  2  P  n  6  2  (2.2.2)  44 with s t r i c t inequality for at least one 6  q  Example 2.2.1 (Stein) Since  E [(c°S -l) P  2  fi  0<(w)s -i) ] p  -  2  (2.2.3) = E {E [(c°S -l) P  6  6  2  - 0KW)S -1) |W]} , P  2  i t i s s u f f i c i e n t (but not necessary) to f i n d a ¥  , such that, for a l l  Ej(c°S -l) 6 P  2  En-measurable function, W  6 > 0,  - (^(W)S -l) |w=w] > 0 P  2  1  a.a.w [P.] o  ,  (2.2.4)  with s t r i c t inequality on a set of p o s i t i v e p r o b a b i l i t y for at least one  For notational s i m p l i c i t y we introduce a random variable j o i n t density, with  f  T f W  T , whose  W , is  ( t , w|6) = K t  2 p  f  s > w  ( t , w|6)  (n+k+4 2)/2- (k-2)/2 -t(1+w)/2-6/2 = Kt w e v  Pr  (6tw) *i=  0  -  1  i!'t(2i+k-2)/2]I  4  D)  t, w > 0  1  2 (with respect to Lebesgue measure on IR ) . Also, a function w i l l be c a l l e d ( s t r i c t l y ) bowl-shaped i f there exists  x^ >  f:(0, °°) ->- IR 0  such that  45 f  i s ( s t r i c t l y ) decreasing on  ( 0 , x^] and ( s t r i c t l y ) increasing on  [x , ) • OT  Q  Then, for fixed  8 and w , E f ( c S - l ) |w=w] i s a s t r i c t l y bowlo P  shaped function of c , which takes i t s minimum at  EjS |w=w] P  - -^ =EJT |w=w] . E [S |W=w] J L  (2.2.6)  P  2p  6  6  It i s easy to show that, f o r each  w  and 6 , f^, ^ ( t j | <$) * ^x^W^' w  i s a non-decreasing function of t , so that sup E [T~ |W=w] = E [T |W=w] 6>0 ° P  p  6  (l+w) 2  But  r[(n+k+2p)/2]  (2.2.7)  §  T([n+k+4p)/2]  P  EQ(T  P  P  |W=W] < c° i n a neighbourhood of the o r i g i n , and therefore,  letting  Y (w) s  = min  { E Q [ T |W=W] P  , c°)",  (2.2.8)  we obtain a procedure which dominates the usual one. d i f f i c u l t to show that, f o r each  Moreover, i t i s not  w , i n f E [T~ |W=w] <_ c° . Therefore, 6>0 P  6  improvement cannot be achieved f o r any w infimum.  by replacing  c° by this  Note that the method of proof used i n this example may be viewed  as a modification of that used i n [46] and [47] .  46  More generally, i f 8 s u f f i c i e n t to f i n d a  E f(c°S -l) o P  i s any sub-a-algebra of  - 0KW)S -1) |8] > 0  2  , then i t i s w ¥ , such that, f o r a l l 6 >. 0 ,  B-measurable function  P  B  a.e. [P ] , o  2  (2.2.9)  with s t r i c t inequality on a set of p o s i t i v e p r o b a b i l i t y f o r at l e a s t one 5  q  .  As i n Example 2.2.1, f o r each atom  E [ ( c S - l ) IB] o  i s a s t r i c t l y bowl-shaped  P  minimum at  E^[T |B]  .  P  B  of  8 , and  function of  The construction of  y  c  6 >. 0 , which takes i t s  now requires that we  determine, f o r each atom  B , whether  from  for which the answer i s a f f i r m a t i v e , f  c° .  For those  B  fined i n the obvious way.  Otherwise  E^(T J B ]  i s bounded away ( i n  P  ¥  i s set equal to  6)  i s de-  c° .  Example 2.2.2. (Brown) If we f i x r > 0  6>0  E  [ ~ | °' T  6  P  Wet  and l e t  r ] ] =  B ^  = 8 ( 1 ^ j(W)) , we see that r  V ~ | e[°» T  P  w  ^  r  < °  '  c  (2.2.10)  We obtain the r e s u l t by l e t t i n g  ( w ) = min {c°, E r T " | B P  Y ( r )  o  V  p  c°  ](w)>  r (n k 2p)/2] / (k-2)/2  ( 1 + u )  r[(n k 4 )/2] 7 u  (l u)-  t  2  ( r )  +  +  +  u  +  P  (  k  -  2  )  /  2  -(n k 2p)/2 +  +  +  (  n  +  k  +  4  p  )  d  /  u  2  w  du  ,  r  .  w > r .  (2-2.11)  47  (2.2.10) can be obtained most e a s i l y by noting that decreasing l i k e l i h o o d r a t i o with respect to  ^  and  ^f^  a r e  (t | S) has  non-  (t|0) , which i n turn  has increasing l i k e l i h o o d r a t i o with respect to  Since  f,j,  f^,(t) .  a n a l y t i c , we know that the corresponding  n o t  r  estimators are inadmissible ([32], [6]).  In f a c t , i f we are searching for  admissible procedures, i t seems reasonable to ask 'a p r i o r i ' why we should r e s t r i c t our search to procedures which are measurable with respect to small  a-algebras (such as  8^).  of proof implied by (2.2.9) .  The answer appears to l i e i n the method  Looking at each atom separately means that  we have l o s t any desirable effects which might accrue from averaging over atoms (see "not necessary" i n Example 2.2.1).  This loss w i l l probably be  most s i g n i f i c a n t i f the number of atoms i s large. observe that the method may too large.  I t i s i n t e r e s t i n g to  f a i l completely i f the  a-algebra chosen i s  For example, i f we do not require the orthogonal invariance,  then we might look f o r a scale-invariant procedure, y(Y)S^ is  8^-measurable.  I f we attempt  to improve on separate atoms of  then the proof f a i l s to produce an improved estimate. we can improve on c e r t a i n atoms of  ,where  8,. M  •  •  ¥ 8^ ,  At the same time, .  * i l ' * "'» I \ l '  In the next section we w i l l construct a minimax, formal Bayes s c a l e invariant estimator, y ^ ' M S ^  .  The construction w i l l be carried out  roughly as follows: a)  s e l e c t a sequence of  a-algebras  B  t B  ,  48  b)  for each c)  B -measurable function, Y , m m  s e l e c t a "good" °  let  m , and  H<*(W) = lim ty (W) . m m-xio  An obvious choice for  i s given by  (c°, E Q [ T ^ |S ] (W)) - but this approach f a i l s because  I* (W) = min  we see that  ty m  m  lim ty (W) = m m-x»  S  (W)  by  In the next section we s h a l l see that  B measurable function i s motivated by a  that our choice of a "good" monotonicity property of  .  -  m  {^( ) -^r>0  * ^  anc  r  v  t  *  desire to avoid  ie  c° (and hence, possibly, to obtain a n a l y L i c i t y ) .  2.3  The construction of a minimax, formal Bayes scale-invariant estimator Although the i n a d m i s s i b i l i t y of  ,  i'^ j(W)S  P  r  considerations, a more d i r e c t proof can be given. 0 < r ' < r , then spect to  f | T  w < r  f | T  W < r  i s clear from a n a l y t i c i t y In f a c t , i f we s e l e c t  , ( t | 0 ) has increasing l i k e l i h o o d r a t i o with re-  ( t | 0 ) , so. that  E tT" |We[0, r']] < E [T~ |w [0, r ] ] , P  0  P  0  and we can therefore repeat the argument i n Example 2.2.2 ty'  truncation  which i s better than  £ [T 0  P  e  ' (w) =<jE [T" |we[0, r ] ] 0  V  c  to f i n d a function  . , where (r)  |w [0, r']] P  Y  ty,  e  0 < w < r' r'<w<r  o w > r  (2.3.1)  49  = min{¥ ,(w) , y (r  n  {r } _ og C X  ct  —  —  ,  be a double sequence of constants such  1  1  {r | B-1 , . . . , V C { r  c)  r an  a3  ( a + 1 )  3 13=1, ... , n ^ } f o r a l l  a  , and  +oo  a  l i m max ^~ -\SQ^ a-xo Kg<n  r -r , J 3 a(3-1) a  a  i s better that  =0  1  Now, f o r each  that  1 p  0 = r . < r < . . . < r au al an ' a  b)  d)  (w)}  oo  More generally, l e t  a)  (r)  .  a , i f we l e t ¥ (w) =  min KB<n  v(w) , then <P ( r  a  aB  }  c° . But i t i s easy to see, with the help of (2.2.11),  ¥ -»•¥*, where a Y*(w) = V. v(w)  (2.3.2)  r[(n k 2p)/2] 7 0 - 2 ) / 2 +  u  +  ( 1 + u )  2Pr[(n-fk 4 )/2] / ( k - 2 ) / 2 +  P  u  -(n lc 2p)/2  ( 1 + u )  +  +  -(n k 4 )/2 +  +  P  d  d  u  u  o  o  2»r[«*Mp)/2J  W<  T  •'  rl  u< " k  2 ) / 2  (l-u)<" P- » + 4  2  2  du  50  Fatou's Lemma enables us to conclude that  Y*(W)S  P  i s minimax, and we  w i l l now show that i t i s also formal Bayes scale-invariant. Consider an a r b i t r a r y nonrandomized >KY)S  , with r i s k function equal to  P  scale-invariant procedure,  f  f (^(y)s -l) ,P  2  f  c v  ( s , yl\)dy ds .  IR (R +  k  Suppose that we are given a (possibly improper) p r i o r measure, Ji , on IR p Among rules of the form  y(Y)S  , the (formal) Bayes rule with respect to  p II  i s Y^WS  , where, f o r each  y , ^(y)  i s  chosen to minimize, and  make f i n i t e , / / (y(y)s -l) + k IR R P  +  f  2  k  ( s , y|A)dnU)ds ' v  .  (2.3.3)  Therefore, i f  g (s, y) = / n  f_ ( s , y|x)dn(A) ,  (2.3.4)  v  k  9  1  H- (y) = [/ s g ( s , y)ds] [/ s ^ g ( s , y ) d s ] - . 0 o P  n  P  n  _J  n  (2.3.5)  p That  Y*(W)S  i s formal Bayes w i l l be established i f we produce a  p r i o r , n , such that g (s, y)K(y) = f .(s|0) S|W<||yl| TT  11  2  (n+k-2)/2 -s/2 !l = Ks e J ,,y  v  2 k/2-1 -sw/2 w e dw ,  (2.3.6)  51  where  K:  .  v  k (R  (0, »j i s a measurable function.  (n+k-2)/2  g (s, y) = Ks v  -s/2 .  e  -||s y-X|l /2 11 1/2  / e  "  2  But, .  (2.3.7)  dll(A) .  2 d £ ( x ) = e HH ^  Therefore, l e t t i n g for  £  dn(X) , and  2  x = s~  y, we are searching  such that  / e ' k IR A  and  A  x  1 — -1 2 dc(X) = K / ( l - v ) e o 2  X v  /  2  dv ,  (2.3.8)  i t follows that  d£(X)  1 ~ -1 — 2 = /(l-v) v e""All o 2  2  dv dX .  / 2 V  In other words, the density of  (2.3.9)  II , with respect to k-dimensional Lebesgue  measure, i s given by CO TI(X)  If  —1  -—  = / u  2  (u+1)"  2  e"il H A  1  u  /  2  du .  (2.3.10)  k = 1 , then  ir(A) = IXI  1  / v (1+X" v) 2  2  - 1  e~  v / 2  dv ,  so that assumption a4 of Section 1.5 i s s a t i s f i e d with  (2.3.11)  3=1.  Moreover,  a l , a2, and a3 are s a t i s f i e d , and  u(X) < K | x |  _ 1  .  (2.3.12)  52  Therefore, as a consequence of Theorem 1.5.1, we have the following theorem:  Theorem 2.3.1:  If  k = 1 , then  1'*(W)S  ,  P  i s a minimax estimator of. a  2  ,  which i s admissible within the class of s c a l e - i n v a r i a n t procedures.  It i s i n t e r e s t i n g to observe that the method of proof used to construct ty* enables us to demonstrate the minimaxity of a number of other estimators. That i s , i f ¥*(w) _< ^(w) of  .  i s a non-decreasing < c  o  , f o r a l l w , then  function of P  VMS  w , and i f  i s a minimax estimator of  Moreover, i f we wish to obtain a minimax, admissible  using t h i s method, then we should also have  ^(w)  <_ ^(w)  , for  estimator all w .  P Otherwise, we could demonstrate the i n a d m i s s i b i l i t y of Example 2.2.1.  Also notice that  as i n  ^ ( O ) = V<j(0) , and f o r a l l  F i n a l l y , i t i s clear that the o r i g i n i s not important blem.  Therefore, i f  E, i s an a r b i t r a r y vector i n P  timator possessing properties s i m i l a r to f*(W)S  P  J (W)S /A  f o  i n the pro-  k IR , then an es-  i s given by  , where  Y*(W)  and  1  w  = YMW^)  (2.3.13)  = S ||X-5|| . _1  2  p Moreover, "i'*(W)S  has an obvious i n t e r p r e t a t i o n . F i r s t notice that o  the natural (minimax and admissible) estimator for  a  i n the analogous  53  problem with known mean  V(S, X;  O  E, i s given by  r[(n+k+2p)/2]  =  (S+||X-f )  P  .  (2.3.14)  2 r[(n+k+4p)/2] P  In the problem with unknown mean estimate of  y , and use estimator  f i n d , i n f a c t , that  X = £  y , we now l e t ¥*(W)S  P  K represent a preliminary  to estimate  2p cr .  I f we  (supporting our p r i o r suspicion), then our  estimate w i l l agree with that given by  Y(S, X; ?) . Otherwise, the  estimate i s modified, depending on the (normalized) distance,  S"^^!x--cj| , between estimate approaches  X  and  K . As this distance becomes i n f i n i t e the  cS  This i n t e r p r e t a t i o n i s s i m i l a r to that of Stein's estimator.  In  this case we note that  (2.3.15)  T^CW )S f  u = £  may thus be regarded as the r e s u l t of f i r s t testing the hypothesis at a p a r t i c u l a r s i g n i f i c a n c e l e v e l - using  hypothesis i s accepted, and 2.4  ^(S, x; 0  i f the  c°S^ i f the hypothesis i s rejected.  Extension of the previous results to G-invariant loss functions Although we assumed squared error loss i n the preceding sections,  the s p e c i f i c form of the loss function played only a minor role i n the proofs.  The form was important only i s so f a r as i t produced  the bowl-  shaped nature of c e r t a i n r i s k functions, and also because i t enabled us  54  to use  the observation that a s t o c h a s t i c a l l y larger  estimate f o r  .  Thus, i f we  a r b i t r a r y , nonegative, 6-invariant F = {fg |  ( I <$) s  w < r  6 >. 0  :  and  L(a; u ,  assume  S  a) = L(aa  loss function, and  0 < r <^ <»}  produces a smaller 2 p  )  i s an  i f we l e t  , then we have a c t u a l l y proved  the following more general r e s u l t :  Theorem If  2.4.1 (i)  ffnr o r ao il ll  f f „e  FP , h (c) = / L ( c s ) f ( s ) d s P  is a strictly  o bowl-shaped function of  (ii)  minimum at  c(f) ,  f^, f  and  2  e F  f-  f L  c  which assumes i t s ( f i n i t e )  increasing,  1 2  implies  c(f ) < c(f ) , 1  and  ( i i i ) Y°(w)  <i' (W)S  then  0  2  = c(f | s  w < w  (.|0))  i s an estimator of  a  ,  which i s formal Bayes s c a l e - i n -  v a r i a n t with respect to the p r i o r given i n (2.3.10), and which has a r i s k function which i s uniformly no larger than that of the best  G-invariant  estimator. The minimaxity of the usual estimator, and hence of often follow from r e s u l t s of K i e f e r and Kudo.  ¥°(W)S  Note that the  of Fatou's Lemma prevents the strenthening of the "no  , will  application  larger than" conclu-  sion i n the theorem, without a more c a r e f u l analysis of the r i s k or a l t e r n a t i v e l y the demonstration of s c a l e - a d m i s s i b i l i t y . of course, that the new  P  functions,  We would hope,  estimator w i l l often be scale-admissible,  particu-  l a r l y i f the t a i l s of the loss function are steeper than those f o r squared error.  55  I t i s perhaps useful to observe that the assumptions of Theorem 2 . 4 . 1 are s a t i s f i e d i n a few important cases: Example 2 . 4 . 1 If  I^Cx) = | x - l |  , then  ¥°(w) = median (U~ |W<w) P  Q  (2.4.1) = median~ (u|w<w) P  where  f  y w  ( u , w|6)  = Ku  P  f  g w  ( u , w|6)  Example 2 . 4 . 2 Brown has shown that the use of an unbiased estimator i s e s s e n t i a l l y equivalent to the use of  Y°(w) = E ^(S |wiw) P  L (x) = x - 1 - ln(x) .  In this case  (2.4.2)  .  Example 2 . 4 . 3 Another common loss i s given by 2 L ( x ) = In (x) , for which 3  y°(w) = e x p { - E ( ^ S*|w<w) • Q  (2.4.3)  56  In Chapter'3 we s h a l l also apply Theorem 2.4.1 i n an i n t e r v a l estimation problem.  2.5  The exponential d i s t r i b u t i o n with unknown location and scale It i s evident that the techniques of the previous sections may be use-  f u l i n many location-scale problems.  In this section we s h a l l examine the  a p p l i c a b i l i t y of the method i n a somewhat a n t i t h e t i c a l s i t u a t i o n . X^, X^,  we s h a l l assume that  X  n  Namely,  are independent observations from  an exponential d i s t r i b u t i o n with density  a N (x) = /  -1  -(x-u)/a e  x >_ u (2.5.1)  (  V 0  x < u .  The problem at hand i s the estimation of a (p>0) , where the loss function P  i s given by  . -2p, p.2 L(a; u, a) = a (a-a )  In this case, (M, X) i s a s u f f i c i e n t s t a t i s t i c , where and  nX = X, + ... + X . For convenience, l e t 1 n  M =  min X_^ , i=l,...,n  S = X - M . Again the  problem remains invariant under the l o c a t i o n - s c a l e group,  . Here, the best  G^-invariant estimator coincides with the maximum l i k e l i h o o d estimator, o P and i s given by 0 _ 0.1 E  ( S  c S , where p'  .  n r(n+p-l) P  ( 2  5  2 )  57  If we l e t form  ¥(Y)S  Y = MS  .  P  , then any s c a l e - i n v a r i a n t estimator i s of the  In analogy with Stein's estimator for the variance of a  normal population, Arnold [1] and Zidek [47] have produced a s c a l e - i n v a r i a n t estimator, y ^ O O S ^ , which dominates the usual one.  m  n  o  r {  C  ( l + y ) n r(n+p) » r(n+2p) P  I t i s given by  _  P  (2.5.3) y < 0 .  To obtain this estimator using the method of Section 2.2, 1  observe that, for each X =ya bowl-shaped function of  E  X  [ S P  '  -  Y = y ]  2 y , E [(cS -l) |Y=y]  and  P  A  first  is a strictly  c , which takes i t s minimum at  E [T- |Y=y] .  (2.5.4)  p  x  E.[S |Y=y] 2p  Here, T  i s a random v a r i a b l e , whose j o i n t density, with  f . ( t , y|x) T  = Kt  Y  2  P P  f  Y , is  ( t , y|x) S Y "' • ''  X g  y  VI  y|A  (2.5.5) 2p+n-l  -nt-n(ty-X)  v =  Now,  K  t  for  .  6  y >. 0 , f  J  _  T  (0,co)( > t  J  (X,co)  •  (t,y|x) f "'" .(t,y|0) i s a non-decreasing ,  T  _ >.  ft  ( t y )  T  v  function of  t , so that  sup E [T |Y=y] = E [T |Y=y] -co<x<» _P  A  _P  u  (2.5.6)  58  r(n+2p)  But there exists  K  such that  E [T~ |Y=y] < c° , i f P  0  the desired estimator  0 <_ y <_ K , and  i s obtained.  It i s natural to attempt to duplicate the procedure i n Example 2.2.2. Unfortunately,  B^  i f we l e t  = B(J|-_  r  r  j( ))  > then the desired monoton-  Y  However, i f we l e t B' ^  i c i t y properties do not hold.  =  r]  '  then  sup -oo<X<  E [ T " | Y e [ 0 , r ] ] = E [T F  x  F  Q  |Y [0,r]] e  < c  (2.5.7) can be seen most e a s i l y by noting that decreasing  .  (2.5.7)  co  ^x|0<Y<r^ ^  l i k e l i h o o d r a t i o with respect to f | Q T  t  n a s  n o n  ~  ( | 0 ) , which i n turn f c  < Y < r  has increasing l i k e l i h o o d r a t i o with respect to f (t) .  We obtain a  dominating procedure by l e t t i n g A n i n { c ° , E [T  P  Q  |8j ](Y)}  Y > 0  r )  Y < 0 (2.5.8)  c°n-(l+r)- - 1  —  n  1  0  [l-(l r)- n  +  Lc°  y  p+1  L  2 p + 1  ]  0n < Y < r ~ " Y < 0 o r Y > r  59  -n-p+1 -n-2p+l  U-U+Y) o  Y > 0  (2.5.9)  ] Y < 0  c  Unfortunately, this procedure i s not formal Bayes In f a c t , we are no longer c e r t a i n that i t dominates can be no worse.  scale-invariant.  c°S  P  However, i n view of an example of Sacks  - although i t [32] regarding  the exponential d i s t r i b u t i o n , perhaps we are asking too much of a candidate for a d m i s s i b i l i t y when we ask that i t be formal Bayes. F i n a l l y , with the aid of a computer, the r i s k function of has been plotted.  ¥*(Y)S  P  From the r e s u l t s , i t i s apparent that the procedure  dominates the usual one, and, i n f a c t does s i g n i f i c a n t l y better over a wide range of values f o r X .  60  Chapter 3:  Minimax, Inadmissible Interval Estimators Of Scale Parameters  3.1 Minimax Estimators In Location-Scale  Problems  Valand [40] has given conditions under which a best l o c a t i o n - i n v a r i a n t i n t e r v a l estimator of a l o c a t i o n parameter w i l l be minimax, when t h i s i s the only unknown parameter i n the problem.  By applying a suitable  transformation,  Valand e a s i l y extends these r e s u l t s to include the i n t e r v a l estimation of scale parameters.  In t h i s section we obtain s i m i l a r r e s u l t s for problems  involving both unknown l o c a t i o n and scale parameters. of the problem, the action space i s unspecified  In our  formulation  (aside from invariant s t r u c t u r e ) ,  and the r e s u l t s are therefore applicable i n both point and i n t e r v a l estimation problems.  In p a r t i c u l a r , i n the next section Lemma 3.1.1  i s used to prove  the minimaxity of c e r t a i n i n t e r v a l estimators of scale parameters, when the l o c a t i o n i s unknown. As we have mentioned i n previous chapters Kiefer  [22] and Kudo [23]  have given s u f f i c i e n t conditions for the minimaxity of best invariant estimators i n a wide range of problems.  In p a r t i c u l a r , t h e i r r e s u l t s are applicable  i n many problems involving unknown l o c a t i o n and scale parameters.  In f a c t ,  applying Kiefer's Theorem i n our problem, we obtain a r e s u l t which i s s i m i l a r i n form to Theorem 3.1.1, but i n which the loss function - not the region of integration - has been truncated.  Kudo's Theorem i s widely  applicable, but to apply i t i n p a r t i c u l a r problems i s d i f f i c u l t .  We,  like  Valand [40], obtain e x p l i c i t conditions by a simple d i r e c t argument.  More  recently, Chen [8] has studied location-scale problems i n which the action space and parameter space coincide. In our formulation X = (U, V, W) ,  of the problem, we observe a random v a r i a b l e  taking values i n  X = U x \J x W ,  61  where  If = (-<*>, ») , 1/ = (0, ) , and (W, C) 00  space.  Assume that the conditional d i s t r i b u t i o n of  i s of the form 0 = { ( u , a):  T?(r--^- , ~|w)  -oo<u<co  a>o}  5  .  unknown. Also, assume that G  on  W .  yj-y H(  a  i s an a r b i t r a r y measurable  If  (Y , Y„, 1 2  above model.  W  (y, a)  i s an element of  i s assumed to be known, but  and  a  are  i s d i s t r i b u t e d according to a known measure i s an observation from  ) , then a suitable transformation w i l l y i e l d the  a  Moreover,  W  may not appear i f the problem has been reduced  Note that the r e s u l t s which follow can e a s i l y be extended  to the higher dimensional location-scale problem i n which .  y  n  , ' *" ,  by s u f f i c i e n c y .  p eI  W = w ,  y -y  2  a  F  Y ) n  n  y -y ,  , where  (U, V) , given  li = IR  and  However, for s i m p l i c i t y of notation, we have treated the case  k = 1 . The action space,  A , and loss function,  L: A x 0 -> [0, ) , are 00  unspecified, but we assume that there exists a homomorphic image, of the location-scale group,  G , which acts on  A  i n such a manner that  G .  A  i s equipped with a  the problem remains invariant under a-algebra,  6 0 , A)  and  [c, d]  and  G , respectively.  [c, d]  6: X x A  A , f o r each  Denote a t y p i c a l element of  G  The class of rules a v a i l a b l e , [0, 1] , such that  A e A , and ( i i ) <5(x, •)  i s measurable, for each  p r o b a b i l i t y d i s t r i b u t i o n on  and l e t  Here,  A , which contains a l l singletons.  V , consists of a l l possible functions (i)  G ,  by  x e X . [c, d] , c > 0  and  - °° < d <  represent i t s homomorphic images i n  Then  [c, d] (u, v, w) = (cu+d, cv, w)  is a  G  00  ,  62  [c, d] (u, a) = (cu+d, ca)  (3.1.1)  [c, d] ' [c', d'] = [ c c % cd'+d] and  [c, d] ^ = [c \ t : X -> A  If  -de  .  i s any nonrandomized invariant r u l e , then i t i s easy  to see that t ( u , v, w) = [vT*u]<j>(w) , f o r some measurable function  (3.1.2)  A . The r i s k function of any such  <(>: W  procedure i s constant, and i s equal to  r(<j>; p, a) =  r(<j>;  0 ,  1)  = / // L([Ou] <Kw); W tixl/  0 ,  l ) d F ( u , v|w)dG(w).  Let D represent the set of a l l measurable functions that there e x i s t s  r(<> j ;  0 , 1)  0  (3.1.3)  <f>: W -»• A , and assume  < t > e D , such that o '  = i n f r(4>; <|>eD •  0 , 1)  Denote the quantity i n equation  < »  .  (3.1.4)  by  (3.1.4)  R  Q  .  OO  Finally, l e t ^ ^ ^ = 1 ^  )f  U x V , such that  ea  n  W  K  n  U x V = Q N=l  g M>) = / / / L([v, u]Kw); w  i  0 ,  c  r  e  a  s  i 8 sequence of compact subsets n  , and define  g : D -* [ 0 , °°) by  l ) d F ( u , v|w)dG(w) .  N  We are now i n a p o s i t i o n to state the main r e s u l t of t h i s section.  (3.1.5)  63  Theorem 3.1.1:  The best invariant r u l e ,  {V, U] <J>(W) , o  i s minimax,  providing that l i m i n f g (cf>) = i n f l i m g (<f>) . N-x» <j>eD <j>eD N-*» Note:  (3.1.6)  The left-hand side i s no greater than the right-hand side, and  the right-hand side i s equal to  R  by the monotone convergence theorem.  q  l i m i n f g (<j>)> R N+~ <j>eD  Therefore, (3.1.6) i s equivalent to  Proof:  .  I t i s well-known that a rule with constant r i s k i s minimax, providing  that i t i s e-Bayes, f o r a l l e > 0  (see, f o r example, [14], p.91).  As  the best invariant rule i s formal Bayes with respect to the measure induced on  0  by r i g h t Haar measure on  G , (see, f o r example, [43]), i t i s natural  to look at p r i o r s which approximate t h i s measure.  x (y, a) = C  where  '•^J-'MFI  C  M  If  1  =  '  /  1  ;  s  o"  a  s  e  j r (y, a) ,  1  c  To t h i s end, l e t  l  u  e  n  c  e  C3.1.7)  °f compacts subsets of  0 , and  ^ T -  C 3  - 1  8 )  t : X -> A i s an a r b i t r a r y nonrandomized procedure, then i t s Bayes  r i s k with respect to  RCt, T ) = C m  = C  M  M  i s given by  / / / / / L [ t C u , v, w); y , c ] d F ( ^ , J | ) d G ( w ) ^ W Uxl/ o o  A  w  (3.1.9)  M  / / / / / L[t(y+au, A W Uxl/ M  a v , w); y , a ] d F ( u ,  v| ) w  d G  ( )dM£ w  (3.1.10)  64  = C  M  / / f/f  I L[t(y+au, cv, w); y, a ] ^ 4 d F ( u , v|w)dG(w) . (3.1.11)  Now, for each (u, v , w) , l e t brackets,  so that  R(t,  T )  (x, y) = [a, y](u, v)  i n the i n t e g r a l i n  i s equal to  (3.1.12) f f f/f  f  w uxv^z where  But  L [ t ( x , y, w); x-v  1  yu, v  - 1  cu, v)  B (u, v) = {[c,  [v, u] [y, x]  v|w)dG(w) ,  yJ  d] Cu, v)  M  y]^4dF(u,  (x-v  yu, v  , Cd, O e A ^  -1 y) = (0, 1) , and therefore,  because of  the assumed invariance of the l o s s ,  R(t, T ) = C f f f/f n  W Uxl/^B  n  f M  L([vT^] [y'Tx]" t(x, y, w); 0, 1)^-}  .  :L  (u, v)  7  J  • dF(u, v|w)dGCw)  C3.1.13) (3.1.14)  - C  M  M  where  f f/f  0 W l  E M  f f . LCIvT'u] Cx, y)  E  [ y T ^ J t C x , y, w); 0, DdFCu, v | w ) d G C w ) ) ^ , -1  J  m  ( x , y) = {[c, d]  -1 (x, y)  y  , (d, c ) e ^ } .  In order to prove the theorem, i t i s s u f f i c i e n t to show that, f o r a l l e > 0 , there exists  M  such that  R(t, x ) > R M o w  - e , f o r a l l nonrandomized t  65  Assume, f o r the present, that there exist subsets  {R^  } '  (- , "0 * CO, )  and  {H^} N=l  of  (i)  H^j i s open and  00  D-HJJ  M  M >  (iii)  CO  N, M=l  n  such that  + (-», <*>)' x (0, «>)  E ( x , y)  (ii)  and  00  .  , for a l l  N ,  N  lim C  / /  *~  V><  Now, given  = 1 , for a l l N . 7  e > 0 , choose  for a l l < j > e D , and l e t  N  N. , such that  be such that  i s j u s t i f i e d by the compactness  g^ (<j)) 3  > R  Q  -  /2 ,  . This l a t t e r step  of  Then, f o r a l l M > N , C3.1.15) RCt,  T  ) > C  /  S If  VM  w  J S  L([Cu]  tyrx] t(x, _ 1  «N  y, w; 0, l)dF(u, v|w)dG(wV  „ „ dxdy  5  y  > (R - /2) C / o M £  /  y  .  (3.1.16)  and the r e s u l t follows. It remains to show that conditions  ( i ) , ( i i ) , and ( i i i ) .  » {H\^} , and {R^ ^ } Let  exist, satisfying  66  2 M  r \  =  I ~  e  >  2 M , 3 *  e  -M  r  U  M, ,  e  J  ,  -1 M C,, = 4 e . M, and M 2  so that  2 / \ -M M I -1 | M. , y) = {(u, v ) : ye <v<ye , |uv y-x|<e } .  v / t^Cx,  r  N  Also, l e t 2 /  =  M  M  ~  ( e  2 M  '  6  M  ~  6  e  \  v / M  ) < x  N e  _  XT"  M  .  N  e  1  M  \  )»  so that  ;  ;  dxdx = [ e 4  M 2  - e ] [M - log(N)] . M  -1 For  N > 2 , the conditions then hold with  Usually,  < J >  take, f o r each  = {(u, v ) : N "<v<N, -Nv<u<Nv} J  i s determined c o n d i t i o n a l l y on W . That i s , we may  w , <|>(w) to be that value of a o  which minimizes,  and makes f i n i t e ,  /  /  L Q v T u J a ;  UxV  0, l)dF(u, v|w) ,  providing that the < j > , so constructed, Let  R ( ) represent W  Q  C3.1.17) i s measurable.  the minimum value i n (3.1.17), and  h : A x W ->- [0, ») be defined by  67  h C a , w) = / / L Q v , u]a; 0, l)dF(u, v|w) N  .  (3.1.18)  Theorem 3.1.2: If ( i )  l i m i n f h^Ca, w) = i n f l i m h^Ca, w) , f o r a l l w , N-*» aeA aeA N-*°-  and ( i i )  Proof:  i n f h^Ca, w) aeA  i s G-measurable,  converges monotonically to  by the monotone convergence /  inf aeA-  Now,  for a l l N , then (3.1.6) holds.  As i n Theorem 3.1.1, we can prove using the hypotheses that  i n f h^Ca, w) aeA  W  (3.1.19)  R (w) , for a l l w . Therefore, °  theorem,  h fa, w)dG(w) -*• / R (w)dG(w) =R W °  N  given  e > 0 , i f we choose  /  i n f h ( a , w)dG(w) > R  W  aeA  .  C3.1.20)  °  N  such that  - e ,  (3.1.21)  °  N  then, for a l l ty e D ,  / h. C<Kw), w)dG(w) > R - e . W ° T  C3.1.22)  N  g 0f>) = / \ (ty(w), w)dGCw) , and  But  W  W  N  the proof i s complete. Now,  assume that  V  i s an a r b i t r a r y topological space, and that  CO  {g } N=l  i s a non-decreasing sequence of non-negative functions on  V  68  Also assume that inf  lim  g  (y) =  S  < »» .  (3.1.23)  Whether we are looking at the unconditional problem, as i n Theorem or at a conditional problem, as i n Theorem  3.1.1,  3.1.2,  we require  conditions under which lim i n f N-**> yeV  g Cy) = N  S  .  (3.1.24)  The following lemma i s p a r t i c u l a r l y useful to that end i n i n t e r v a l estimation problems. Lemma  I f g^ i s lower semicontinuous for a l l N , then  3.1.1:  holds, i f and only i f , for any and  e > 0 , there e x i s t s a compact set  K , such that g„ (y) > S o N o o  Proof:  K,  - e , for a l l y £ K. . ' J  The proof i n one d i r e c t i o n i s t r i v i a l , and we s h a l l use contradiction  to prove the other d i r e c t i o n . hold, and l e t  and N o  S - lim i n f ° N yeV  as above.  Therefore, assume that  (3.1.24)  g,,(y) = e > 0 . Using t h i s  does not  e , choose  K  N  Then, for a l l N > N , there exists - o  y„ e K N  such that  g ( y ) = i n f g (y) y^K  C3.1.26)  < S  C3.1.27)  N  N  N  q  - £ .  Since  K  If  i s f i x e d , i t follows from the lower semicontinuity  N  (3.1.24)  i s compact, the sequence  there e x i s t s a subsequence  {y^ } i  {y }' has an accumulation point y N • o  such that  of g  N  that  69  (3.1.28)  But from the monotoriicity of  {g } > i t follows that M  (3.1.29) N.  i  l  x  We.arrive at a contradiction by taking the l i m i t . For completeness, the following easy lemma i s stated without proof. Lemma 3.1.2; If  ( i ) L(»;0, 1) for a l l w ,  then (3.1.6) holds.  i s bounded, or i f  ( i i ) F(-,'|w)  has compact support,  703.2 Interval Estimation Of Scale Parameters  consider the problem of determining a confidence i n t e r v a l f o r a Usually, a confidence c o e f f i c i e n t  1-a  i s selected,  and the confidence i n t e r v a l i s assumed to be of the form  (p>0) .  0 < a < 1 , (c^ S ,  S ) .  P  P  Then c., i = 1, 2 , are chosen so that  (3.2.1)  where  d  distribution.  Equation (3.2.1) does not uniquely determine the  c_^ .  So some a d d i t i o n a l c r i t e r i o n i s introduced which, with (3.2.1), an i n t e r v a l . (i)  determines  Common c r i t e r i a , so introduced are:  shortest i n t e r v a l of c o e f f i c i e n t  1-a: minimize  °2 ~ ]_ ' c  subject to (3.2.1). (ii)  l o g a r i t h m i c a l l y shortest i n t e r v a l of c o e f f i c i e n t tnic^)  minimize (iii)  - In(c^)  , subject to (3.2.1).  equal-tailed i n t e r v a l of c o e f f i c i e n t 1-d: select such that  d /  1-a:  « f„(s)ds = / f ,(s)ds  d^ and d^  = a/2 .  In order to obtain the form of the shortest or l o g a r i t h m i c a l l y shortest confidence i n t e r v a l s , i t w i l l be useful to present a simple measure-theoretic lemma.  Assume  (a)  (X, M)  i s a measurable space,  (b)  y  and  are cr-finite measures on  V  (X, M)  absolutely continuous with respect to (c)  f  (d)  g = f * of A  (e)  y £  (f) h ( A ) = k  and  (X, M ) ,  denotes the Radon-Nikodym d e r i v a t il v e  with respect to  e M , and  y is  v ,  i s a non-negative measurable function on 5 where  such that  v ,  g "*"[(c, " O J ^ A ^ C g "*"[[c, °°) ] , f o r some  v ( A ) - /fdy , A e M , where k  i s any fixed  c > 0 , scalar.  A  Lemma 3.2.1: If v ( A ) < v ( A ) , A E H , then  (a)  /fdy < / fdy . A  C  c v(A)> v(A) .  /fdy > / fdy , A E M , then  (b) _ i f  A  A  A  c (c)  h ( A * )  = inf  Proof:  (a) Since  Therefore,  h(A)  v(A) < v ( A) , v(A ~ A ) < v ( A ~ A ) . c c c  /fdy = /gdv < A-A  c  cv(A~A  )  <  cv(A~A)  °  A~A  c  and the r e s u l t follows. by noting that  .  AEM  K  °  < / gdv = /fdy , A  C  ~A  A ~A  C  The proof of (b) i s s i m i l a r , and we obtain (c)  h ( A ) = /[k -g]dv . A  It w i l l be convenient to introduce some notation. Definition:  (a) If f : iR-> IR  f(x ) = f(x ) . (b) For x > 0 x  , then  x^ =  ff] i f  2  and  -  00  < a <  00  , Q (x) = x  ct  a  n  d  only i f  72  Now,  c_ - c. = p / d  V (A) = /x A  x  Therefore,  i f we l e t X = (0, ) , 00  l  dx , and  1  dx .  y = Lebesgue measure, we see that  y (A) =  /x^*^ A  dv (x) , so that 1  Lemma 3.2.1  = Q, , 1+p  dy dv.  .  Since Q_ , • f 1+P S  i s unimodal, i t follows from  (a) (b), that the shortest confidence  i s given uniquely  i n t e r v a l of s i z e  1 - a  by  2 / f (s)ds = 1 - a d  (3.2.2) and  d  ±  = d  2  [Q  .f ]  1 + p  g  Solutions to (3.2.2) may  be obtained  Similarly, letting  from [24]  v.. (A) = / x "'"dx A  s  and  [39].  we see that the l o g a r i t h m i c a l l y  1  shortest i n t e r v a l of size  1 - a  i s given uniquely  by  d 2  /  f_(s)ds = 1 - a  "i and  d  ±  (3.2.3) = d  2  [Q  1  -f]  This l a t t e r i n t e r v a l i s also "shortest unbiased" (see, for example, [33]), and solutions, to (3.2.3) may  be obtained  from [24], [28], and  [39].  From a decision-theoretic point of view, what properties do the above procedures possess?  In i n t e r v a l estimation problems, two  types of losses occur.  quite d i f f e r e n t  One varies with the " s i z e " of the i n t e r v a l .  other i s a consequence of not covering the true parameter. valued loss function seems appropriate. and a d m i s s i b i l i t y can then be developed.  The  Thus, a vector-  Extended d e f i n i t i o n s of minimaxity A l t e r n a t i v e l y , by taking a l i n e a r  73  combination of the i n d i v i d u a l losses, a real-valued loss function may be constructed.  For a discussion of the r e s u l t i n g implications, see, f o r example,  Blythe [ 3 ] . Let  A = {(a^, a^)0  < a^<a2<°°}  denote the action space, and assume  that the l o s s function i s of the form  L(a^, a^\ V, cr) = L(a^a , a^o ) 2 p  As i n Section 2.1, the problem i s then invariant under  2p  .  G , and any G-invariant  procedure i s of the form (c^S , c^ S ) . P  P  In p a r t i c u l a r , i f L C«, 3 ) = k C3-ct) + 1 - 1  (1), (a,S)  (3.2.4)  then, from Lemma 3.2.1, the shortest i n t e r v a l of c o e f f i c i e n t n  best G-invariant procedure, for some L Ca, 8) = k*£n(Ba ) 1  1  k  *  > 0 .  1-a  i s the  Similarly, i f  + 1 - I (1), (a,3)  (3.2.5)  the l o g a r i t h m i c a l l y shortest i n t e r v a l i s best G-invariant. It does not appear to have been generally recognized that the equalt a i l s i n t e r v a l i s also best G-invariant, when  CO  + /  Now, keeping i s given by  £n(  s )f (s)ds . P  C;L  c_c  s  -1  f i x e d , the d e r i v a t i v e of  r with respect to c  74 C  -1/P 2  OO  c ^ ! - / f (s)ds + /  f (s)ds] ,  1  C  (3.2.7)  s  -I/P  and the r e s u l t i s evident.  With t h i s smoother loss function, the s t a t i s t i c i a n  i s not merely assessed a unit loss whenever interval. a  '  a  i s not contained  2 p  i n the  Instead, he i s penalized according to the v -distance between 2  and the i n t e r v a l .  2 p  We w i l l now  use the r e s u l t s of Section 3.1. to show that these best  G-invariant procedures are minimax.  In the notation of the previous section,  1/2 replace  U  and  V  by  be i d e n t i f i e d with = [-N, N ]  K  X  A .  x [N  and  S  , respectively,  D  may  In order to apply Theorem 3.1.1, l e t  , N J , so that  - 1  and note that  g : A -> [0, ~)  ±  N  defined by  s  N  g (a N  a) =  l S  2  [*(N)  -  * ( - N ) ]  k  (a ~a 2  a v )f (v)dv .  f^Ua^ *,  2 p  2  2  N  Using l o s s function =  K  L^,  we see that  ) •/'_ v f (v)dv + P r ( N < V < N ) 2p  1  • [ $ (N)-<j> ( - N ) ]  g (a^, a ) N  -  - 1  v  v  2  / f (v)dv  -k  .  [N~ VN]n[a ~ ,a '" ] 2p  1  2  Moreover, i t i s evident that  R- < 1 .  2p  1  Therefore, using Lemma 3.1.1  and  the continuity of g^ , i t i s s u f f i c i e n t to show that there e x i s t s a compact subset (a , ±  K  a) 2  of  N  A , and  t K .  Let  N  q  q  , such that  be such that  g^. (a^, a^) > 1 - e ,' for a l l [$(N )-$(-N Q  In order to f i n d the form of the compact set, l e t  ) ]  k  > (1-e) ^ 1  M.. > 0  2  .  be such that N  Pr(V > M~ ) 2p  > (l-e)  1 / 2  .  Moreover, l e t  M  2  satisfy  k* M  2  / ° N  o  v f ( v ) d v > 1. 2 p  v  75  Finally, let K =  , a„):  {(3  0 < a  Ca^, a^) ft K,  Then, f o r  <  a  either  1  ~ i a  2  ••  M } 2  a^ >  , or  a  2  - a^ > M  .  2  In the f i r s t  case, i t follows that a" k  g  >  Ca , a ) 1  N  2  [$(N )-$C-N )J o  o  K  2 p  1  [1- /  o In the second case,  f (v)dv] > 1 - e . v  o ^ N  %  C  2  l'  a  a  >  )  I  "  V  H  Using loss function  g. Ca , N  a„)  1  £*(N)-*(-N)]  -k  $  C-N )J k*(a -a ) / o  k  2  0  1  v f (v)dv > ( l - e ) 2 p  v  1 / 2  > 1-e  L^,  =  * -1 k -cnCa^ ) 1  N  f  N  f (v)dv + P r ( N v  -1 O  X  <V<N ) Q  - 1  -/f Cv)dv v  [N ,N ]n[a ,a ] 1  2p  o  o  Again,  k  * 1  2  R  2p  1  < 1 .  A s i m i l a r argument w i l l y i e l d the r e s u l t , where M  0  M  M  /  2  N  f ( y ) d v •> 1 , and v  _  K = { ( a ^ a ) : 0 < a^ < 2  , a  2  > e  2  2  satisfies  .  1  0  The proof of the minimaxity of the e q u a l - t a i l s i n t e r v a l i s easier, since L„(a , a ) -> 3 ni n2 n  of  00  , whenever  (a , , a „) n l ' n2  approaches the one-point compactif i c a t i ion r r r r  A . Using the r e s u l t s of Chapter 2,  i t i s now possible to demonstrate the  i n a d m i s s i b i l i t y of the l o g a r i t h m i c a l l y shortest and e q u a l - t a i l e d i n t e r v a l s . One would suspect that the shortest i n t e r v a l would also be but the method seems hard to apply i n t h i s case.  inadmissible,  76  F i r s t consider a l o g a r i t h m i c a l l y shortest i n t e r v a l -l/p c^ o  where  o  -l/p IQ-^*fg] •  Let  we s h a l l show that there exists an (c° YCWjS^, c°'l'CW)S )  (but the same v -volume).  P  1  .  In t h i s case,  H-invariant procedure  Note that an a r b i t r a r y H-invariant procedure  2  (H^OOS *, V (W)S )  i s of the form  ln(y)  q  P  having uniformly higher p r o b a b i l i t y of coverage  P  As we  _ y = c^ • c^ o  =  (c°S , c°S ) ,  P  1  2  s h a l l be r e s t r i c t i n g ourselves to i n t e r v a l s having v^-volume A * = {(ac°, ac°): 0 <a<  , let  we are now  »} .  A*  Identifying  with  (0,  00  ) ,  i n a p o s i t i o n to apply Theorem 2.4.1.  But, i n order to apply Theorem 2.4.1, a stronger version of Lemma 3.2.1  i s needed.  It i s necessary to e s t a b l i s h that the minimization  t h i s lemma i s "monotone", when the relevant integrand particular,  if  g^)  < g(a )  , gO^)  2  < g(b )  2 / fdy < l  2  , then  2  < b  1  2 I fdu . l  a  v[(a^, a ) ] = v[(b^, b ) ]  , a  2  i s unimodal.  in In  , and  b  a  b  A simple geometric  (or calculus) argument w i l l show that t h i s i s the case, providing that X  there e x i s t s  q  such that  and s t r i c t l y decreasing  g  i s s t r i c t l y increasing on  (0, X  Q  J ,  [ X , «*>) .  on  Q  Therefore, using the notation of Theorem 2.4.1, for a l l f e F , CO  h^(c) =  on  A.  / L (cc^ s , cc s )f(s)ds o P  2  P  2  i s a s t r i c t l y bowl-shaped function  which takes i t s minimum when  Also, i f  f^ f  1 2  i s increasing, and  (cc°) "^  =  d 2  P  = (cc°)  ^2^  '  t  h  e  [Q^*  n  f] .  77  d  2  Cd )  h  f  2  Cd )/f Cd )  L  2  2  2  ^ X• l  d  l  f  (  d  l  )  l Cd )/f Cd )  f  2  1  2  Therefore, using the stronger version of Lemma 3.2.1 again, condition ( i i ) of Theorem 2.4.1 holds. Cc° Y % ) ) ~  1  /  = ( c ¥°(w))"  P  ,0  1/p  2  (c° 1 (W)S , c° Y°(W)S )  then  2  .  [Q • f | 2  g  (:|0)] ,  w < w  (3.2.8)  i s a minimax i n t e r v a l estimator of o"  P  P  with respect to L of  Finally, i f  2p  Its r i s k function i s uniformly ho larger than that  (c°S , c°S ) , and i t i s formal Bayes within the class of s c a l e P  P  invariant procedures i n a d m i s s i b i l i t y of limit.  having v^volume (c°S , c°S ) P  Incidentally,  P  since  £n (y) . Naturally, the actual  would be obtained p r i o r to taking the  ¥ (W) < 1 , the new procedure  produces  shorter i n t e r v a l s . ( c ° S , c S ) . Again, we  Now, consider an equal-tailed i n t e r v a l consider only procedures  of the form  P  P  2  (c° 4 (W)S , c ;  P  (W)S ) P  2  . Here,  the s t r i c t l y bowl-shaped nature of h^(c) i s a consequence of (3.2.7), and the "dominating" Pr  procedure (c°¥°(W)S , c V ° ( W ) S ) P  t S < (c°T°(w))" |w<w] 1/p  6 = Q  =Pr  [ S > (c^ (w))" 0  6 = ( )  P  2  1/p  |w<w] .  satisfies  78  Bibliography Arnold, B. C. "Inadmissibility of the usual scale estimate f o r a s h i f t e d exponential d i s t r i b u t i o n . " J . Amer. S t a t i s t . Assoc., Vol. 65 (1970), pp. 1260-1264. Berk, R. H. "A special group structure and equivariant estimation." Ann. Math. S t a t i s t . , V o l . 38 (1967), pp. 1436-1445. Blyth, C. " A d m i s s i b i l i t y of minimax procedures." Vol. 22 (1951), pp. 22-42.  Ann. Math. S t a t i s t . ,  Brown, L. D. "On the a d m i s s i b i l i t y of invariant estimators of one or more l o c a t i o n parameters." Ann. Math. S t a t i s t . , V o l . 37 (1966), pp. 1087-1135. Brown, L. D. 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